  We present new results on realtime alternating, private alternating, and
quantum alternating automaton models. Firstly, we show that the emptiness
problem for alternating one-counter automata on unary alphabets is undecidable.
Then, we present two equivalent definitions of realtime private alternating
finite automata (PAFAs). We show that the emptiness problem is undecidable for
PAFAs. Furthermore, PAFAs can recognize some nonregular unary languages,
including the unary squares language, which seems to be difficult even for some
classical counter automata with two-way input. Regarding quantum finite
automata (QFAs), we show that the emptiness problem is undecidable both for
universal QFAs on general alphabets, and for alternating QFAs with two
alternations on unary alphabets. On the other hand, the same problem is
decidable for nondeterministic QFAs on general alphabets. We also show that the
unary squares language is recognized by alternating QFAs with two alternations.
