Two problems that might help develop intuition:

1. Prove that $\mathsf{AM} \subseteq \mathsf{PCP}(poly(n),1)$. You are allowed to assume $\mathsf{NP} \subseteq \mathsf{PCP}(poly(n),1)$.

2. Prove that $\mathsf{PSPACE} \subseteq \mathsf{PCP}(poly(n),poly(n))$. Hint:

Use IP = PSPACE

Trivia: The first exercise subsumes the PCP for GNI, while the second subsumes the PCP for permanent. Both results are subsumed by $\mathsf{NEXP} = \mathsf{PCP}(poly(n),1)$.

Two problems that might help develop intuition:

1. Prove that $\mathsf{AM} \subseteq \mathsf{PCP}(\text{poly}(n),1)$. You are allowed to assume $\mathsf{NP} \subseteq \mathsf{PCP}(\text{poly}(n),1)$.

2. Prove that $\mathsf{PSPACE} \subseteq \mathsf{PCP}(\text{poly}(n),\text{poly}(n))$. Hint:

Use IP = PSPACE

Trivia: The first exercise subsumes the PCP for GNI, while the second subsumes the PCP for permanent. Both results are subsumed by $\mathsf{NEXP} = \mathsf{PCP}(\text{poly}(n),1)$.