The complexity of computing the permanent of a $l\times l$ binary matrix is known to be $\#\mathsf{P}$-complete, from the famous result of Valiant, where $l = \Theta(n)$.
We know that the problem is not in $\mathsf{P}$, unless $l = \mathcal{O}(\log_2 n)$. What can we say about the complexity of the problem when $l = \mathcal{o}(n)$ as well as $\log_2 n = \mathcal{o}(l)$ ? For example, if $l = \text{polylog}(n)$ ?
Does the problem remain $\#\mathsf{P}$-complete ? Or is it complete for a class in some level of the polynomial hierarchy ?