Here's some suggestions. I'm going to use a notation like 'Tier 3' or 'Tier 5' to state relatively how advanced I think these ideas are, in terms of actually manifesting as technology - where Tier 0 would ostensibly be 'stuff we have now'.
Physics techs
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Programmable Matter (Tier 1) - By making use of materials that are microscopically designed to constantly be on the cusp of a large number of potential conformational changes but which are entropically 'frustrated', it becomes possible to use specific weak signals to 'toggle' matter at a microscopic level between different sorts of material properties - metals that can turn reflective or matte, conductive or insulating, etc in response to specific patterns of radio signals.
Moire Interferometers (Tier 1) - Interferometry uses the fact that one can fake having a huge receiving dish by having an array of synchronized smaller receivers that are spread out over a large distance. Moire Interferometers invert this idea, using a huge array of far-off sources all being distorted by a single object in order to very precisely image that object. It turns out this can also be used in microscopy, imaging things at higher resolutions than were previously accessible with visible light.
Macroscopic Entanglement (Tier 2) - As an offshoot of failed experiments in quantum computers, researchers managed to create a literal analog of Schroedinger's Cat - a macroscopic object which could be in a stable undecided superposition of states. The result was, even if it couldn't be used for computation, it could be used to make really weird forms of matter that act sort of like 'solid illusions' - being present, or not, as constrained by a carefully controlled wave function collapse at a remote location.
Singular Pinch Focusing (Tier 2) - While mechanical 'chirp' singularities have been known for some time, the observation that colliding black holes could also produce the same divergent timescales led to the invention of a particular type of focusing device. By creating an unstable 'neck' of distorted spacetime and allowing it to collapse, material stored within that neck is converted to energy at very high efficiencies and is squirted out in an extremely tight, collimated beam. Unfortunately, the energy required to create and collapse the neck is usually higher than what is extracted from the contained material, and so this makes for better weapons technology than energy sources.
Mantle Tickler (Tier 2) - Initially the idea was to make measurements of the lower mantle by suspending a probe in a ball of molten iron and letting it sleet through the layers of the planet. After this caused a reflexive release of a mantle plume (instant volcano), the primary purpose of this technology changed significantly.
Causatons (Tier 3) - Investigations into temporal paradoxes in quantum gravity led to the discovery of a new kind of particle/wave which mediates non-local causal relationships. These 'causatons' can be used suppress the probabilities of unwanted events by connecting those events to a microscopic temporal paradox. Causaliton emitters are rated for a certain degree of 'improbability', which is the maximally improbable event they can make happen. Exceeding that threshold tends to cause the operator to have a brain aneurysm just before they press the 'on' button.
Algorithmic Embodiment Theory (Tier 5) - A formalism which carefully explores the practical consequences of the idea that anything for which an equation can be written exists somewhere simply because its existence is precisely implied by that equation. By positing very specific field theories whose existence elsewhere would by implication affect all other universes, it becomes possible to exert pressures on the laws of physics themselves, at least in a limited area.
Informatics/Computational techs
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Informatic Flow Control (Tier 1) - By understanding that information itself is as real a quantity as energy or matter, it becomes possible to redesign complex systems in terms of the flow of information through the system rather than via detailed modelling of each interaction. This permits large-scale blind manipulation of complex systems at the macro scale, so long as you have control of the 'boundaries' of the system. Game mechanical ideas basically center around improving economy/society functions or managing dysfunctions.
High Dimensional Projections (Tier 1) - You'd think that working in a billion dimensions would be harder than working in 3, but it turns out that due to a quirk of mathematics that's not always the case. Through clever projections, the behavior of unpredictable systems can be unfolded and simplified in abstract, infinite-dimensional spaces. The immediate effect is an improvement in military targetting/defenses, but applications to psychological modelling lead towards the first primitive AIs.
Mechanical Turk Inversion Theorem (Tier 2) - By reversing the question of whether a thorough database is identical to an actual intelligence, it became possible to make highly accurate emulators of an individual's behavior. These emulators become unstable in their behavior and decay after a certain number of interactions unless they are constantly reset, so they are not useful as general purpose intelligences. However, they can be constructed from observations and thus provide a method for performing complex what-if scenarios about the behavior of alien civilizations (diplomacy effects?). There are some shady uses as well, making a clone-persona of someone to interrogate it without them knowing (espionage effects?).
Nonlinear Ghost Resonance (Tier 2) - Non-linear systems resonate with each-other most strongly when their parameters are similar. The effect of this resonance flows both ways, allowing systems to not just be measured and predicted, but to be manipulated. This effect can be used to synchronize a sufficiently general informatic system (the 'ghost') with an external system, and then by tuning the local system cause the external system to shift in its behavior. Useful for all sorts of hacking/remote take-over/economic warfare/etc purposes.
Memetic Avatars (Tier 2) - Although extending theories of multiscale lifeforms to models of ideas has been in vogue, it has produced little in the way of workable technology until now. This breakthrough enabled the collective emergent minds of a civilization to be embodied in the form of a person-scale avatar and interacted with directly. While this was very distracting for researchers when it came to talking to the lolcat hivemind, that loss in productivity was more than made up for when the researchers realized they could imbue sentience into their theories and collaborate with them directly.
Engramatic Imprintor (Tier 3) - A realization of the core components of mind enabled the transmission of personae through short periods of interaction combined with the right sorts of feedback calculations and predictive algorithms. Big espionage effect, as spies of this civilization can literally talk their way into someone else's head. Probably hugely destabilizing to society, but might reduce impact of health on productivity (dying minds can leap to a new body).
Recursively Adaptive Society (Tier 3) - By chaining a series of delayed societal models together, it becomes possible to create a society that behaves the way it would behave given slightly longer to experience the consequences of its choices. Iterating this procedure results in a new form of societal organization that instantly adapts to the consequences of its own internal organization (though unpredictable external sources are still difficult to deal with).
Quasi-stable Halting Limit Theory (Tier 4) - Through a new closure on the theory of quasi-finite groups, it becomes possible to tell absolutely whether a given continuous signal will cycle, asymptote, or continue to develop new behavior - incidentally solving the halting problem for a large subclass of continuous computational machines, that is to say, brains. By determining whether a given citizen will continue to have new ideas or simply recycle what they have previously come up with, it becomes possible to allocate labor much more efficiently, and bump stuck individuals into new endeavors (or to retirement).
Ascension Equation (Tier 5) - Chasing the idea of being able to maximally compress the total evolutionary, philosophical, and scientific advancements of a civilization into the smallest possible form, it was discovered that the problem is intrinsically asymmetric - that is to say, generating the future of scientific progress in the present is as hard as actually making that progress, whereas verifying it is exponentially easier. However, buried within this was a worrisome discovery: based on partial verification, it was possible to tell that certain civilizations were simply doomed and were categorically incapable of advancing beyond some finite limit. With knowledge of the so-called Ascension Equation, it became possible to try to detect trends in one's civilization which would lead to a such inevitable dead ends, and to pre-emptively interfere.
Biology
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Metabolic Damping (Tier 1) - The discovery of this particular form of radiation was an unexpected offshoot of longevity research, an attempt to slow metabolism even further during sleep periods and so limit metabolic stresses on the body by interfering with the transfer of electrons within the body's cells by effectively pinning them in place near their nuclei. While the induced state doesn't actually prevent aging, it does make a person very, very inert, and can be reversed safely (most of the time) - making it useful for crowd control and military actions, as well as suspended animation.
Eco-collapse Theory (Tier 1) - An understanding of the stability and instability of ecological networks has led to a method for adding extra interactions between members of the ecosystem in order to render it self-stabilizing and more compact. Among other things, this increases a civilization's resilience against environmental damage, as well as the efficiency of their food production (if they tend to eat things that are higher up in the food chain, at least).
Bonsai Buildings (Tier 1) - Discovery of a bacterium which could be induced to produce carbon nanotubes led to a very strange period in architecture. With this technology, it is possible to lay out a vat of builder microbes and then stimulate them into printing the building's architecture layer by layer, much like a 3D printer.
Neural Repurposing (Tier 2) - Brains are good at specializing themselves towards frequently performed tasks, healing around damage, etc, but the timescales involved prevented that from being any more than a curiosity until now. By drastically increasing neural plasticity, it becomes possible for a person's brain to repurpose itself almost entirely towards a given task. While initially it was thought that this would cause the person to forget almost everything else they'd ever learned, it turns out that holographic residues of those previous memories remained and could be reconstructed later with the right medicines and treatments.
Binary Parasites (Tier 2) - A result of studying opportunistic infections and multi-virus interactions, this technology makes it possible to engineer biological weapons which have a very high specificity and require exposure to two distinct factors in order to take effect, drastically lessening the danger they pose to the civilization which deploys them. A fairly 'scary' tech, diplomatically, I'd think - a civilization researching this is effectively expressing their intent to use biological weapons.
Developmental Regression Chambers (Tier 2) - A medical treatment which fools the body into thinking that it has re-entered embryonic development. This can be used to regrow lost limbs or, with additional gene therapy, change a person's entire body architecture (growing them a second heart, etc).
Auto-Evolution (Tier 3) - As a result of studying the immune system and its evolutionary properties, it became possible to extend those evolutionary dynamics to a person's entire body by implanting a few synthetic organs. The result is that recipients of the treatment adapt very quickly to anything which is killing their cells - heat, cold, hard radiation, etc. Such adaptations come with an increased metabolic cost and overall shorter lifespan, however.
'Weird' techs
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Dual Harmonization Pump (Tier 1) - Physicists of other civilizations call this 'a miracle of cross-scale coupling between phonon fields and the nuclear weak force' or give other similar explanations, but the fact remains that this civilization discovered that a particular kind sound could alter the nuclear decay rates of certain unstable ceramics. The result is that they have to sing to their starship engines to make them go.
Rotated Decomposition of Metric Excitations (Tier 3) - Normally space and time have a nice 3-and-1 relationship that makes life easy. This technology is a practical application of the Wick rotation - a mapping between temperature and imaginary time in quantum field theory. In this case, a field is set up which causes one spatial dimension to be rotated into a second time axis, generally giving all involved a terrible headache. Few species have the mental architecture to handle two perpendicular time directions. Aside from being very disorienting, the rotation causes very little in the way of physical stresses or damage.
The Anthropic Eschaton Principle (Tier 2) - A philosophical idea that combines the anthropic principle with the idea of static time. While the anthropic principle constrains the laws of physics to permit the existence of sentient life to observe the universe in the present, this philosophy states that the choice of moment is irrelevant to the argument, and so the anthropic principle means that the conditions of the universe must be such that sentient life is in existence during all real moments, up to the universe's own ending. This philosophy has a tendency to exaggerate existing societal trends - an optimistic society becomes more optimistic (we'll be alive at the end of the universe, hooray!), whereas a pessimistic society becomes more pessimistic (this means that the universe is going to end some time within our lifetimes...). Civilizations can cause unrest in their neighbors by injecting this idea into their popular media at the right time.
Probability Distortion Telescopes (Tier 1) - This idea was hatched by going off of the theory that hyper-advanced civilizations will have access to time travel. If that is the cause, causal self-consistency should create probabilistic anomalies within the timeline of anyone who will, at some point in the future, interact with that civilization. This led to the development of detective devices which constantly search for correlations in quantum-random signals as evidence for future interaction with causal manipulation. Can be used for funky signal processing, maybe revealing map squares of other civilizations that have developed Tier 3 techs?
