coNP ⊆ AM collapses the polynomial hierarchy

Unconditional · against free reductions · BHZ87, BM88

Statement

If coNP is contained in AM, then ${\mathbf{\Sigma }}_2^{\mathbf{P}}={\mathbf{\Pi }}_2^{\mathbf{P}}=\mathbf{AM}$, so the polynomial hierarchy collapses to its second level — BHZ87; BM88 derive it from the collapse theorem $\mathbf{AM}[k]=\mathbf{AM}$. Whether $\mathbf{coNP}\subseteq \mathbf{AM}$ holds is open.

Sketch

${\mathbf{\Sigma }}_2^{\mathbf{P}}=\exists \cdot \mathbf{coNP}$. If $\mathbf{coNP}\subseteq \mathbf{AM}$, this is contained in $\exists \cdot \mathbf{AM}=\mathbf{MAM}$, which equals $\mathbf{AM}$ by the collapse theorem, and $\mathbf{AM}\subseteq {\mathbf{\Pi }}_2^{\mathbf{P}}$. Hence ${\mathbf{\Sigma }}_2^{\mathbf{P}}\subseteq {\mathbf{\Pi }}_2^{\mathbf{P}}$, which forces equality.