No-Signaling Proofs with $O(\sqrt{\log n})$ Provers are in PSPACE

No-signaling proofs, motivated by quantum computation, have found applications in cryptography and hardness of approximation. An important open problem is characterizing the power of no-signaling proofs. It is known that 2-prover no-signaling proofs are characterized by PSPACE, and that no-signaling proofs with $poly(n)$-provers are characterized by EXP. However, the power of $k$-prover no-signaling proofs, for $20$), then the corresponding 2-prover game has value at most $1 - 2^{dk^2}$ (for some constant~$d>0$). In the second route we show that the value of a sub-no-signaling game can be approximated in space that is polynomial in the communication complexity and exponential in the number of provers.