# Complexity class without fixed-poly size circuit

$$PP$$ 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 $$MA/1$$.