Moments in families of L-functions over function fields

2. Question by Craig Westerland on cohomology coming from quantum shuffle algebras

As racks give braided vector spaces, which give quantum shuffle algebras whose cohomology agrees with that of the Hurwitz space associated to $c$, $\operatorname{Hur}^c_n$, if we consder braided vector spaces whose quantum shuffle algebra has well-understood cohomology (for example, involving Nichols algebras) do we get arithmetic applications?

One preprequisite to this question would be whether we can descend the relevant local systems to $\mathbb F_q$.

Basically, someone has computed some relevant results in cohomology, and this question is asking whether there are relevant arithmetic statistical questions. There are many applications of Nichols algebras of diagonal type by people including Androskiewitsch, Angiono, Witherspoon, etc.