We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let $\mathrm{C}$ be a class of polynomial-size concepts, and suppose that $\mathrm{C}$ can be PAC-learned with membership queries under the uniform distribution with error $1/2 - \gamma$ by a time $T$ quantum algorithm. ... more >>>

A canonical communication problem ${\rm Search}(\phi)$ is defined for every unsatisfiable CNF $\phi$: an assignment to the variables of $\phi$ is distributed among the communicating parties, they are to find a clause of $\phi$ falsified by this assignment. Lower bounds on the randomized $k$-party communication complexity of ${\rm Search}(\phi)$ in ... more >>>

Attempts to prove the intractability of the Minimum Circuit Size Problem (MCSP) date as far back as the 1950s and are well-motivated by connections to cryptography, learning theory, and average-case complexity. In this work, we make progress, on two fronts, towards showing MCSP is intractable under worst-case assumptions.

While ... more >>>