  The Aaronson-Ambainis conjecture (Theory of Computing '14) says that every
low-degree bounded polynomial on the Boolean hypercube has an influential
variable. This conjecture, if true, would imply that the acceptance probability
of every $d$-query quantum algorithm can be well-approximated almost everywhere
(i.e., on almost all inputs) by a $\mathrm{poly}(d)$-query classical algorithm.
We prove a special case of the conjecture: in every completely bounded
degree-$d$ block-multilinear form with constant variance, there always exists a
variable with influence at least $1/\mathrm{poly}(d)$. In a certain sense, such
polynomials characterize the acceptance probability of quantum query
algorithms, as shown by Arunachalam, Bri\"et and Palazuelos (SICOMP '19). As a
corollary we obtain efficient classical almost-everywhere simulation for a
particular class of quantum algorithms that includes for instance $k$-fold
Forrelation. Our main technical result relies on connections to free
probability theory.
