Aaronson–Ambainis influential-variable conjecture

Prove that every polynomial p from {0,1}^N to [0,1] of degree at most t has a variable i whose influence satisfies Inf_i(p) ≥ poly(Var(p)/t).

Background

Aaronson and Ambainis reduced the simulation conjecture to a statement about bounded low-degree polynomials. The conjecture seeks a polynomial lower bound, in terms of the variance and degree, on the influence of at least one variable.

The paper states that despite almost two decades of effort, the best known bound remains exponential in the degree rather than polynomial. The authors use this conjecture only as an implication source for their query-weight conjecture; they do not prove it.

References

In, Aaronson and Ambainis reduced the simulation conjecture to a basic question about low-degree polynomials: Let p : \zoN \to [0,1] be a bounded degree-t polynomial. There is a variable i\in [N] such that \Inf_i(p) \ge \poly\left(\frac{\Var(p)}{t}\right).

— Quantum Speedups Require Structure or Depth  (2608.19158 - Blanc et al., 19 Aug 2026) in Section 1, paragraph titled “The Aaronson--Ambainis Conjecture”

We conjecture that there must then be a “heavy variable”, one for which \mcA allocates a lot of its query weight:

— Quantum Speedups Require Structure or Depth  (2608.19158 - Blanc et al., 19 Aug 2026) in Section 2, subsection “A new conjecture,” Conjecture “Query-efficient quantum algorithms have heavy variables”

As mentioned in the introduction, one can also consider a strong version of~\Cref{conj:our conjecture} with a \polylog(1/\delta) dependence, which we now state formally:

— Quantum Speedups Require Structure or Depth  (2608.19158 - Blanc et al., 19 Aug 2026) in Section 8, subsection “Implications of the regularity lemma,” Conjecture “Strong version of Query-efficient quantum algorithms have heavy variables”

Despite several efforts by experts in the fields of quantum query complexity and Fourier analysis, the Aaronson--Ambainis conjecture remains wide open and it is only known to hold in particular cases .

— Optimal inequalities for completely bounded polynomials and the limitations of quantum query algorithms  (2609.05201 - Gutiérrez et al., 4 Sep 2026) in Section 1, subsection “Aaronson--Ambainis conjecture”