An optimal separation of randomized and Quantum query complexity

We prove that for every decision tree, the absolute values of the Fourier coefficients of given order t≥1 sum to at most (cd/t)t/2(1+logn)(t−1)/2, where 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 t, becoming trivial already at t=√d.