Some of the most interesting and important results concerning quantum finite automata arethose showing that they can recognize certain languages with (much) less resources thancorresponding classical finite automata. This paper shows three results of such a typethat are stronger in some sense than other ones because (a) they deal with models ofquantum finite automata with very little quantumness (so-called semi-quantum one- andtwo-way finite automata); (b) differences, even comparing with probabilistic classicalautomata, are bigger than expected; (c) a trade-off between the number of classical andquantum basis states needed is demonstrated in one case and (d) languages (or the promiseproblem) used to show main results are very simple and often explored ones in automatatheory or in communication complexity, with seemingly little structure that could beutilized.
