We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function F$:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be F$_k:\{0,1\}^n \rightarrow \{0,2^{k-1}\}$ by $F_k(x) = 2^{k-F(x)}$ (mod $2^k$). We consider the fractional agreement (which we refer to as $\gamma_{d,k}(F)$) of $F_k$ with ... more >>>

We study the computational complexity of emptiness problems for circuits over sets of natural numbers with the operations union, intersection, complement, addition, and multiplication. For most settings of allowed operations we precisely characterize the complexity in terms of completeness for classes like NL, NP, and PSPACE. The case where intersection, ... more >>>

For every constant $\epsilon>0$, we give an $\exp(\tilde{O}(\sqrt{n}))$-time algorithm for the $1$ vs $1-\epsilon$ Best Separable State (BSS) problem of distinguishing, given an $n^2\times n^2$ matrix $M$ corresponding to a quantum measurement, between the case that there is a separable (i.e., non-entangled) state $\rho$ that $M$ accepts with probability $1$, ... more >>>