Given snapshot and boolean circuit how to compute coNP formula?
You want to know how, given a circuit $C$, one can express the statement "given $i$, $C$ correctly computes $z_i$, the i'th snapshot in the computation of $M$ on some fixed input $x$", by a coNP predicate. Note that if this can be done, say using the predicate $\varphi (C,x)$, then 
$$\psi(x)=\exists C: \varphi(C,x)\land C\left(2^{|x|^c}\right) \text{ is accepting}$$

is exactly the $\Sigma_2$ formula you're looking for. To see why, note that $x\in L$ iff the last snapshot is accepting, or equivalently if $C\left(2^{|x|^c}\right)$ is accepting, and $C(i)$ correctly computes $z_i$. So if $\varphi(C,x)$ indeed expresses "C correctly computes $M$'s snapshots on input $x$", then $x\in L \iff \psi(x)$. If you have a guarantee that polynomial circuits for $L$ exist, then you can bound the existential quantifier, and only look for poly-size circuits.

So how do we write $\varphi(C,x)$ as a coNP predicate? Using locality, you know that the $i'th$ snapshot depends on some $k$ previous ones (where $k$ is a constant depending on the number of tapes). $C$ correctly computes the snapshots if for all $i,i_1,...,i_k$ (this gives you the universal quantifier) it holds that
$T\left(x, C(i), C(i_1),...,C(i_k)\right)$, where $T$ depends on $M$'s transition function. Intuitively, $T$ checks whether $i_1,...,i_k$ are the previous indicies of snapshots who determine $z_i$, and if so verifies that the transition was indeed according to $M$'s rules.