Aaronson–Ambainis conjecture on influences of Boolean functions

Let $f:\{\pm 1\}^{n}\to[-1,1]$ be a function of degree $d$. Define its variance and the influence of coordinate $j$ by

$$\operatorname{Var}(f):=\sum_{S\neq\emptyset}\widehat{f}(S)^2,$$ $$\operatorname{Inf}_{j}(f):=\sum_{j\in S}\widehat{f}(S)^2.$$

Aaronson–Ambainis conjecture. Every such function satisfies

$$\operatorname{poly}(\operatorname{Var}(f)/d)\leq\max_{1\leq j\leq n}\operatorname{Inf}_{j}(f).$$

This conjecture predicts that a bounded low-degree Boolean function with non-negligible variance has a coordinate of correspondingly non-negligible influence. The source notes that only a very special case was solved using multilinear Bohnenblust–Hille estimates; the general conjecture remains open.