Let $\phi_1$ and $\phi_2$ be the following statements:
$\phi_1:$ There is a function $f:\{0,1\}^*\to\{0,1\}$ computable in $E$ that has circuit complexity $2^{\Omega(n)}$.
$\phi_2:$ There is a function $f:\{0,1\}^*\to\{0,1\}$ computable in $NE \cap CoNE$ that has $2^{\Omega(n)}$ hardness on average.
Q1. Is there any $\Pi_2$ sentence $\psi$ in the language of arithmetic such that we can prove $\mathbb{N}\models \psi \leftrightarrow \phi_1$?
Q2. Is there any $\Pi_2$ sentence $\psi$ in the language of arithmetic such that such that we can prove $\mathbb{N}\models \psi \leftrightarrow \phi_2$?
Actually, I want to know if these conjectures are $\Pi_2$ expressible like $P\not = NP$.