Given polynomials $f,g,h\,\in \mathbb{F}[x_1,\ldots,x_n]$ such that $f=g/h$, where both $g$ and $h$ are computable by arithmetic circuits of size $s$, we show that $f$ can be computed by a circuit of size $\poly(s,\deg(h))$. This solves a special case of division elimination for high-degree circuits (Kaltofen'87 \& WACT'16). The result ... more >>>

We show that given a quantum measurement, for an overwhelming majority of pure states, no meaningful information is produced. This is independent of the number of outcomes of the quantum measurement. Due to conservation inequalities, such random noise cannot be processed into coherent data.

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We prove a SOS degree lower bound for the planted clique problem on Erd{\"o}s-R\'enyi random graphs $G(n,1/2)$. The bound we get is degree $d=\Omega(\epsilon^2\log n/\log\log n)$ for clique size $\omega=n^{1/2-\epsilon}$, which is almost tight. This improves the result of \cite{barak2019nearly} on the ``soft'' version of the problem, where the family ... more >>>