  In this paper we explore the power of AM for the case that verifiers are {\em
two-way finite automata with quantum and classical states} (2QCFA)--introduced
by Ambainis and Watrous in 2002--and the communications are classical. It is of
interest to consider AM with such "semi-quantum" verifiers because they use
only limited quantum resources. Our main result is that such Quantum
Arthur-Merlin proof systems (QAM(2QCFA)) with polynomial expected running time
are more powerful than in the case verifiers are two-way probabilistic finite
automata (AM(2PFA)) with polynomial expected running time. Moreover, we prove
that there is a language which can be recognized by an exponential expected
running time QAM(2QCFA), but can not be recognized by any AM(2PFA), and that
the NP-complete language $L_{knapsack}$ can also be recognized by a QAM(2QCFA)
working only on quantum pure states using unitary operators.
