Interactive Oracle Proofs (IOPs) form the backbone of some of the most efficient general-purpose cryptographic proof-systems. In an IOP, the prover can interact with the verifier over multiple rounds, where in each round the prover sends a long message, from which the verifier only queries a few symbols.

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Time-bounded conditional Kolmogorov complexity of a string $x$ given $y$, $K^t(x\mid y)$, is the length of a shortest program that, given $y$, prints $x$ within $t$ steps. The Chain Rule for conditional $K^t$ with error $e$ is the following hypothesis: there is a constant $c\in\mathbb{N}$ such that, for any strings ... more >>>

We prove that Boolean matrices with bounded $\gamma_2$-norm or bounded normalized trace norm must contain a linear-sized all-ones or all-zeros submatrix, verifying a conjecture of Hambardzumyan, Hatami, and Hatami. We also present further structural results about Boolean matrices of bounded $\gamma_2$-norm and discuss applications in communication complexity, operator theory, spectral ... more >>>