Let
$L = \{ n : \text{the }n^{th}\text{ binary digit of }\pi\text{ is }1 \}$
(where $n$ is thought of as encoded in binary). Then what can we say about the computational complexity of $L$? It's clear that $L\in\mathsf{EXP}$. And if I'm not mistaken, the amazing "BBP-type" algorithms for computing the $n^{th}$ bit of $\pi$ using quasilinear time and $(\log n)^{O(1)}$ memory, without needing to compute the previous bits, yield $L\in\mathsf{PSPACE}$.
Can we do even better, and place $L$ (say) in the counting hierarchy? In the other direction, is there any hardness result whatsoever for $L$ (even an extremely weak one, like $\mathsf{TC}^0$-hardness)?
An interesting related language is
$L' = \{ \langle x,t\rangle : x\text{ occurs as a substring within the first }t\text{ digits of }\pi \}$
(where again, $t$ is written in binary). We have
$L' \in \mathsf{NP}^L$
and hence $L' \in \mathsf{PSPACE}$; I'd be extremely interested if anything better is known.