Analogous characterization for quantum vs randomized query complexity

Determine an analogue of the completion-complexity characterization that rules out superpolynomial quantum speedups in comparison to deterministic query complexity by developing a precise characterization for when the bounded-error quantum query complexity Q(f) is polynomially related to the bounded-error randomized query complexity R(f) for partial Boolean functions f.

Background

The paper proves that completion complexity characterizes when superpolynomial quantum speedups (over deterministic algorithms) are possible: D(f) is polynomially related to a measure M(f) if and only if M is extendable without superpolynomial blowup via a completion (Lemma/Result in the completions section).

An open direction is to obtain an analogous, necessary-and-sufficient criterion for the relationship between quantum and randomized query complexities, i.e., to understand when Q(f) and R(f) are polynomially related for partial functions via a completion-like framework.