Origin of quantum complexity theory
The class BQP (see also Complexity Zoo: BQP) was defined by 


E. Bernstein and U. Vazirani. Quantum complexity theory. SIAM Journal on Computing 26, pp.&nbsp;1411-1473 (1997).


The class QMA (see also Complexity Zoo: QMA) was defined by


A. Kitaev. Quantum NP. Talk at AQIP’99: Second Workshop on Algorithms in Quantum Information Processing, DePaul University (January 1999).


which apparently built on the idea of a "quantum proof" suggested by 


E. Knill. Quantum randomness and nondeterminism. Technical Report LAUR-96-2186, Los Alamos National Laboratory (1996). [arXiv:quant-ph/9610012].


An important relative to QMA, which is perhaps more delicate but which in a certain respect is a more exact analogue of NP for quantum computation, is the class QMA1 (Complexity Zoo), in which the completeness constraint is perfect (i.e.&nbsp;for YES instances, there is a "quantum proof" with acceptance probability 1, albeit for a noiseless quantum computer). This was defined in


S. Bravyi. Efficient algorithm for a quantum analogue of 2-SAT (2006).
[arXiv:quant-ph/0602108]


There are other interesting quantum classes, of course. If you want an introduction to the subject, a good start would be 


John Watrous. Quantum Computational Complexity. Encyclopedia of Complexity and System Science, Springer (2009). [arXiv:0804.3401].