Uncertainty Principle for Quadratic Fourier Analysis

Determine a sharp lower bound on m in representations of the form AND(x) = ∑_{i=1}^m c_i (−1)^{q_i(x)}, where q_i: F2^n → F2 have degree at most 2; in particular, prove or refute that m must be exponential in n (i.e., m ≥ 2^{Ω(n)}).

Background

The question asks how many quadratic phase functions are required to express the indicator of the all-ones vector (the n-ary AND) as a real linear combination. Linear (degree-1) phases require m = 2^n and suffice, but the power of allowing degree-2 phases is unclear.

Connections exist to constant-degree circuit complexity hypotheses, suggesting that exponential lower bounds may hold. Current results provide only linear lower bounds (m ≥ n).