We show that the basic semidefinite programming relaxation value of any constraint satisfaction problem can be computed in NC; that is, in parallel polylogarithmic time and polynomial work. As a complexity-theoretic consequence we get that MIP1$[k,c,s] \subseteq $ PSPACE provided $s/c \leq (.62-o(1))k/2^k$, resolving a question of Austrin, Håstad, and ... more >>>

Suppose we want to minimize a polynomial $p(x) = p(x_1, \dots, x_n)$, subject to some polynomial constraints $q_1(x), \dots, q_m(x) \geq 0$, using the Sum-of-Squares (SOS) SDP hierarachy. Assume we are in the "explicitly bounded" ("Archimedean") case where the constraints include $x_i^2 \leq 1$ for all $1 \leq i \leq ... more >>>

In both query and communication complexity, we give separations between the class NISZK, containing those problems with non-interactive statistical zero knowledge proof systems, and the class UPP, containing those problems with randomized algorithms with unbounded error. These results significantly improve on earlier query separations of Vereschagin [Ver95] and Aaronson [Aar12] ... more >>>