We study low-degree testing for group-valued functions over a Boolean slice. Specifically given a degree parameter $d$ and oracle access to a function $f:\{0,1\}^n_{n/2} \to G$ where $\{0,1\}^n_k$ denotes the set of vectors in $\{0,1\}^n$ of Hamming weight $k$ and $G$ is an Abelian group (not necessarily finite), the low-degree ... more >>>

We give quantitative bounds on the sparsity of a real polynomial with non-negative coefficients vanishing on a slice of the Boolean cube $\{\pm1\}^n$. We obtain similar bounds on the dimension of real orthogonal representations of (the complement of) Johnson-type graphs. The estimates are related to the classical problem about ... more >>>

The quantum analogue of the PCP theorem for QMA remains wide open. A central obstacle is the local indistinguishability of quantum codes: every sufficiently small view of an encoded witness is independent of the witness, seemingly preventing a local verifier from distinguishing YES from NO instances. One approach to this ... more >>>