No-Signaling Proofs with O(√(log n)) Provers are in PSPACE
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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 2<k<poly(n) remained an open problem. We show that k-prover no-signaling proofs (with negligible soundness) for k=O(√(log n)) are contained in PSPACE. We prove this via two different routes that are of independent interest. In both routes we consider a relaxation of no-signaling called sub-no-signaling. Our main technical contribution (which is used in both our proofs) is a reduction showing how to convert any sub-no-signaling strategy with value at least 1-2^-Ω(k^2) into a no-signaling one with value at least 2^-O(k^2). In the first route, we show that the classical prover reduction method for converting k-prover games into 2-prover games carries over to the no-signaling setting with the following loss in soundness: if a k-player game has value less than 2^-ck^2 (for some constant c>0), 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.
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