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