# Quadratically Tight Relations for Randomized Query Complexity

Let f:{0,1}^n →{0,1} be a Boolean function. The certificate complexity C(f) is a complexity measure that is quadratically tight for the zero-error randomized query complexity R_0(f): C(f) ≤ R_0(f) ≤ C(f)^2. In this paper we study a new complexity measure that we call expectational certificate complexity EC(f), which is also a quadratically tight bound on R_0(f): EC(f) ≤ R_0(f) = O(EC(f)^2). We prove that EC(f) ≤ C(f) ≤ EC(f)^2 and show that there is a quadratic separation between the two, thus EC(f) gives a tighter upper bound for R_0(f). The measure is also related to the fractional certificate complexity FC(f) as follows: FC(f) ≤ EC(f) = O(FC(f)^3/2). This also connects to an open question by Aaronson whether FC(f) is a quadratically tight bound for R_0(f), as EC(f) is in fact a relaxation of FC(f). In the second part of the work, we upper bound the distributed query complexity D^μ_ϵ(f) for product distributions μ by the square of the query corruption bound (corr_ϵ(f)) which improves upon a result of Harsha, Jain and Radhakrishnan [2015]. A similar statement for communication complexity is open.

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