Is there a general theorem that tells us when we can hope for exponential speedups from quantum algorithms, and when we cannot? In this paper, we make two advances toward such a theorem, in the black-box model where most quantum algorithms operate.

First, we show that for any problem that is invariant under permuting inputs and outputs and that has sufficiently many outputs (like the collision and element distinctness problems), the quantum query complexity is at least the $7^{\text{th}}$ root of the classical randomized query complexity. (An earlier version of this paper gave the $9^{\text{th}}$ root.) This resolves a conjecture of Watrous from 2002.

Second, inspired by work of O'Donnell et al. (2005) and Dinur et al. (2006), we conjecture that every bounded low-degree polynomial has a “highly influential” variable. (A multivariate polynomial $p$ is said to be bounded if $0\le p(x)\le 1$ for all $x$ in the Boolean cube.) Assuming this conjecture, we show that every $T$-query quantum algorithm can be simulated on most inputs by a $T^{O(1)}$-query classical algorithm, and that one essentially cannot hope to prove $\mathsf{P}\neq\mathsf{BQP}$ relative to a random oracle