In the seminal work of \cite{Babai85}, Babai have introduced \emph{Arthur-Merlin Protocols} and in particular the complexity classes $\MA$ and $\AM$ as randomized extensions of the class $\NP$. While it is easy to see that $\NP \subseteq \MA \subseteq \AM$, it has been a long standing open question whether these classes are actually different.
In \cite{BGM06}, B{\"{o}}hler et al. introduced the probabilistic class of $\SBP$ and showed that $\MA \subseteq \SBP \subseteq \AM$. Indeed, this is the only known natural complexity class that lies between $\MA$ and $\AM$. In this work we study the relations between these classes further, partially answering some open questions posed in \cite{BGM06}.

Title change. Addressing some comments, extending the main result.

In the seminal work of \cite{Babai85}, Babai have introduced \emph{Arthur-Merlin Protocols} and in particular the complexity classes $MA$ and $AM$ as randomized extensions of the class $NP$. While it is easy to see that $NP \subseteq MA \subseteq AM$, it has been a long standing open question whether these classes are actually different.
In \cite{BGM06}, B{\"{o}}hler et al. introduced the probabilistic class of $SBP$ and showed that $MA \subseteq SBP \subseteq AM$. Indeed, this is the only known natural complexity class that lies between $MA$ and $AM$. In this work we show that the class $AM$ collapses to $SBP$ if $UNIQUE$-$SAT$ is $NP$-hard, where $UNIQUE$-$SAT$ stands for the problem of deciding whether a Boolean formula has a unique satisfying assignment or none.