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 = 2k bits defined by a complete binary tree of NAND gates of depth k, which achieves Ro(f) = O(D(f)07537...). 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 Õ(√n). We further show that the quantum query complexity of the same function is Õ(n1/4), 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 gonn variables that has zero-error randomized query complexity Ω(n/log(n)) and bounded-error randomized query complexity R(g) = Õ(√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) = Õ(√n). These functions show that the relations D(f) = O(R1(f)2) and Ro(f) = Õ(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 Ro, a 3/2-power separation between Qe 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 Göos, 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.
This work was funded by the European Commission 7th Framework Programme; grant id erc/615307 - Progress in quantum computing: Algorithms, communication, and applications (QPROGRESS)
Algorithms and Complexity

Ambainis, A, Lee, T. J, Balodis, K. (Kaspars), Santha, M, Belovs, A, & Smotrovs, J. (2016). Separations in query complexity based on pointer functions. doi:10.1145/2897518.2897524