  The purpose of this paper is to prove the upper bound in Malle's conjecture
on the distribution of finite extensions of $\mathbb{F}_q(t)$ with specified
Galois group. As in previous work of Ellenberg-Venkatesh-Westerland, our result
is based upon computations of the homology of braid groups with certain
(exponential) coefficients. However, the approach in this paper is new, relying
on a connection between the cohomology of Hurwitz spaces and the cohomology of
quantum shuffle algebras.
