  Following an early work of Dwork and Stockmeyer on interactive proof systems
whose verifiers are two-way probabilistic finite automata, the authors
initiated in 2004 a study on the computational power of quantum interactive
proof systems whose verifiers are particularly limited to quantum finite
automata. As a follow-up to the authors' early journal publication [J. Comput.
System Sci., vol.75, pp.255-269, 2009], we further investigate the quantum
nature of interactions between provers and verifiers by studying how various
restrictions on quantum interactive proof systems affect the language
recognition power of the proof systems. In particular, we examine three
intriguing restrictions that (i) provers always behave in a classical fashion,
(ii) verifiers always reveal to provers the information on next moves, and
(iii) the number of interactions between provers and verifiers is bounded.
