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We give a general principle that converts fixed-uniformity bounds for forbidden traces into bounds with polynomial dependence on the uniformity. More precisely, let $H$ be a fixed set system on $h\ge1$ vertices, and suppose that, for some $\alpha\ge0$, $\operatorname{ex}_j(m,\operatorname{Tr}(H))=O_{H,j}(m^\alpha)$ for...
We give the first exponential quantum advantage in the general interactive three-party Numbers-on-Forehead (NOF) model for a decision problem. Previous separations hold only for restricted protocols like one-way communication for a relation. We construct an explicit partial Boolean function, the Interleaved Unitary Pro...
We develop an argmax principle for analyzing sum-of-squares relaxations of optimization problems over the unit sphere. Given a feasible pseudo-expectation, we form a polynomial of high-order pseudo-moments, such as $\Phi_k(u)=\widetilde{\mathbb E}\langle x,u\rangle^{2k}$. Our guiding principle is that its maximizers ar...
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