In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function f on n = 2 k bits defined by a complete binary tree of NAND gates of depth k, which achieves R 0 (f ) = O(D(f ) 0.7537... ). We show this is false by giving an example of a total boolean function f on n bits whose deterministic query complexity is Ω(n/ log(n)) while its zero-error randomized query complexity is O( √ n). We further show that the quantum query complexity of the same function is O(n 1/4 ), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities.We also construct a total boolean function g on n variables that has zero-error randomized query complexity Ω(n/ log(n)) and bounded-error randomized query complexity R(g) = O( √ n). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is Q E (g) = O( √ n). These two functions show that the relations D(f ) = O(R 1 (f ) 2 ) and R 0 (f ) = O(R(f ) 2 ) are optimal, up to poly-logarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R 0 , a 3/2-power separation between Q E and R, and a 4th power separation between approximate degree and bounded-error randomized query complexity.All of these examples are variants of a function recently introduced by Göös, Pitassi, and Watson which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.

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We initiate a study of random instances of nonlocal games. We show that quantum strategies are better than classical for almost any 2-player XOR game. More precisely, for large n, the entangled value of a random 2-player XOR game with n questions to every player is at least 1.21... times the classical value, for 1 − o(1) fraction of all 2-player XOR games.

The paper gives a short introduction to mutually unbiased bases and the Welch bounds and demonstrates that the latter is a good technical tool to explore the former. In particular, a criterion for a system of vectors to satisfy the Welch bounds with equality is given and applied for the case of MUBs. This yields a necessary and sufficient condition on a set of orthonormal bases to form a complete system of MUBs.This condition takes an especially elegant form in the case of homogeneous systems of MUBs. We express some known constructions of MUBs in this form. Also it is shown how recently obtained results binding MUBs and some combinatorial structures (such as perfect nonlinear functions and relative difference sets) naturally follow from this criterion.Some directions for proving non-existence results are sketched as well.

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