We discuss some known and introduce some new hierarchies and
reducibilities on regular languages, with the emphasis on the
quantifier-alternation and difference hierarchies of the
quasi-aperiodic languages. The non-collapse of these hierarchies and
decidability of some levels are established. Complete sets in the
levels of the hierarchies under the polylogtime and some
quantifier-free reducibilities are found. Some facts about the
corresponding degree structures are established. As an application,
we characterize the regular languages whose balanced leaf-language
classes are contained in the polynomial hierarchy. For any
discussed reducibility we try to give motivations and open
questions, in a hope to convince the reader that the study of these
reducibilities is interesting for automata theory and computational
complexity.