$PP$ is shown to have no fixed-poly size circuit by Vinodchandran.
Bounded inside the polynomial hierarchy, $\Sigma^2_p$ is also shown to possess no fixed-poly size circuit by Kannan.
In notation, we can write $$ PP\nsubseteq SIZE(n^k)\\ \Sigma^2_p\nsubseteq SIZE(n^k) $$ for all $k$.
Has there been any improvement on these results recently?
We do not take into account non-uniform class like $MA/1$.