PSPACE is contained in IP

By [A1] there is a polynomial-time computable map $f$ with $x\in L$ if and only if $f(x)\in \mathrm{T}\mathrm{Q}\mathrm{B}\mathrm{F}$; since $f$ runs in polynomial time, there is a polynomial $q$ with $\mathrm{\mid }f(x)\mathrm{\mid }\le q(\mathrm{\mid }x\mathrm{\mid })$ for every input $x$.

Define the verifier $V_L$ for $L$ as follows: on input $x$, compute ${\mathrm{\Phi }}_x:=f(x)$ deterministically and then run the TQBF verifier $V$ of [A2] on input ${\mathrm{\Phi }}_x$, forwarding the prover's messages to $V$ and $V$'s messages to the prover. Since $f$ is computable in polynomial time and $V$ runs in time polynomial in $\mathrm{\mid }{\mathrm{\Phi }}_x\mathrm{\mid }\le q(\mathrm{\mid }x\mathrm{\mid })$, the machine $V_L$ is a probabilistic polynomial-time machine, and the protocol has polynomially many rounds and polynomially bounded communication because those bounds for $V$ are polynomial in $\mathrm{\mid }{\mathrm{\Phi }}_x\mathrm{\mid }$.

If $x\in L$ then ${\mathrm{\Phi }}_x\in \mathrm{T}\mathrm{Q}\mathrm{B}\mathrm{F}$ by step 1.1, so by [A2] the honest prover for $V$ on ${\mathrm{\Phi }}_x$, used as the prover for $V_L$, is accepted with probability one, in particular at least $2\mathrm{/}3$.

If $x\mathrm{\notin }L$ then ${\mathrm{\Phi }}_x\mathrm{\notin }\mathrm{T}\mathrm{Q}\mathrm{B}\mathrm{F}$ by step 1.1, so by [A2] every prover for $V$ on input ${\mathrm{\Phi }}_x$ is accepted with probability at most $1\mathrm{/}3$; a prover for $V_L$ on $x$ induces such a prover for $V$ on ${\mathrm{\Phi }}_x$, the reduction being deterministic, so no prover for $V_L$ exceeds $1\mathrm{/}3$.

Steps 1.1 and 2.1 exhibit for $L$ a probabilistic polynomial-time verifier with polynomially bounded interaction, and steps 2.2 and 2.3 give completeness $1\ge 2\mathrm{/}3$ and soundness at most $1\mathrm{/}3$; hence $L\in \mathrm{I}\mathrm{P}$ by [A3]. Since $L$ was an arbitrary language in PSPACE, $\mathrm{P}\mathrm{S}\mathrm{P}\mathrm{A}\mathrm{C}\mathrm{E}\subseteq \mathrm{I}\mathrm{P}$, with perfect completeness for every language in the class.