Why must QMA complete problems be promise problems? I'm reading Watrous's excellent survey paper on paper on quantum complexity theory. In it he states that it would be surprising if a QMA -complete problem were found to have a vacuous promise (I.e. Be a language). Why is this so?

Does it have to do with the fact that the k-local Hamiltonian problem is a promise problem?

Also, this leads me to a related question: are there QMA-complete problems that aren't inherently "quantum" in nature?
On the second question: http://arxiv.org/abs/0905.4755v2 gives a classical QMA-complete eigenvalue problem related Markov chains.