Dhiraj Holden: No-Signaling Proofs with sqrt(log⁡ n) Provers is in PSPACE

Friday, December 6, 2019 - 10:30am to 12:00pm
Hewlett, G882
Dhiraj Holden, MIT

Abstract: 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 = √(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^(-Omega(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.