We prove that for every decision tree, the absolute values of the Fourier coefficients of given order ℓ 1 sum to at most cwhere n is the number of variables, d is the tree depth, and c > 0 is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal (arxiv 2019; FOCS 2020). The bounds prior to our work degraded rapidly with ℓ, becoming trivial already at ℓ = √ d. As an application, we obtain, for any positive integer k, a partial Boolean function on n bits that has bounded-error quantum query complexity at most ⌈k/2⌉ and randomized query complexity Ω(n 1−1/k ). This separation of boundederror quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis (STOC 2015). Prior to our work, the best known separation was polynomially weaker: O(1) versus n 2/3−ε for any ε > 0 (Tal, FOCS 2020).
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