What is the Relationship between QMA and AM? I read in S. P. Jordan, D. Gosset, P. J. Love's "$QMA$-complete problems for stoquastic Hamiltonians and Markov matrices" that it is unlikely that $QMA \subseteq AM$.

I was surprised about this assertion. So what is the proper relationship between $QMA$ and $AM$?
No relationship is known to hold between QMA and AM, and it is reasonable to conjecture they are incomparable.

If QMA were proved to be contained in AM, it would be an absolutely enormous result in quantum complexity.  Of course it would imply that BQP is in PH, which itself would be huge, but it would go beyond that -- it would surely require major revelations about the structure of quantum algorithms and quantum certificates.

Having said that, the evidence against is not very convincing.  An oracle relative to which QMA is not contained in AM would help, and it seems like such a result may not be far off -- but we don't even have this yet.

A proof of the reverse containment, AM in QMA, would also be huge.  At least here we have an oracle relative to which AM is not contained in QMA (and in fact is not even contained in PP).