I know the following:
$$IP[k] \subseteq AM[k+2]$$
Now, I also know that
$$ \\#SAT_D \in IP[poly]$$
(As shown on page 159 of Arora/Barak).
In their proof, (and the following proof of $$ TBQF \in IP$$)
the prover sends the verifier polynomials
the verifier picks random number between 0 & p, $$2^n < p < 2^{n+1}$$
and using linearisation (to keep the polynomial's degree down) and the probabilistic polynomial identity testing, we have an IP system.
So here is my question: in all of these proofs, the verifier NEVER hides his coin tosses from the prover (he just picks random numbers, and sends them to the prover).
So does this in fact mean that PSPACE = AM[poly] ?
Thanks!