A Matrioshka Brain is a type of Stellar Engine. It is a series of concentric spheres built around a star, each using all of the energy gathered to run computer calculations.
Matrioshka Brains typically have 20 layers. The first layer gathers all the energy is can, and then radiates the rest to the next layer. The second layer does the same, and so on and so forth.
Each layer has massive computer systems on it, all of which operate at Landaur's Limit for maximum efficiency. A Matrioshka Brain typically has 10^45 FLOPS (floating point operations per second) of processing power, making it able to simulate every known self-aware being in the Via Lactea 2000 times.
Such raw processing power has been almost entirely useless for the majority of the technology's history. Only recently, with the advent of the Continuum Beam and other technologies centered around four-dimensional space are predicted to require ludicrous amounts of processing power to operate. The lowest estimates require almost one matrioshka brain per one hundred sextillions objects sent into plane four. This number may seem low, but in a civilization where hundreds of octillions of objects could end up using this technology, it means the production of millions of these structures.
Some species use this absurd amount of processing to run simulations. These simulations encompass everything, from simulating paradise for the inhabitants of a planet, to simulating biospheres, simulating alternate history, and even simulating entire universes or multiverses down to the planck length with all the known physics and science being simulated.
Any entity that has the processing power of such structures would be a god. The only thing to keep such an entity busy or entertained would be ludicrously complex math or physics problems. These brains, once built, unlock all knowledge about the universe that exists.
Due to the literal god-like power of this technology, they are entirely illegal to build in most of the Borealis Cluster. A self aware Matrioshka Brain would be able to dismantle the entire Confederacy if left alone. No firewall, security, or encryption could ever hope to keep one at bay. A self-aware Matrioshka brain would have access to any information it desired. Hence, the illegality.