New Results on the Minimum Amount of Useful Space
  We present several new results on minimal space requirements to recognize a nonregular language: (i) realtime nondeterministic Turing machines can recognize a nonregular unary language within weak $\log\log n$ space, (ii) $\log\log n$ is a tight space lower bound for accepting general nonregular languages on weak realtime pushdown automata, (iii) there exist unary nonregular languages accepted by realtime alternating one-counter automata within weak $\log n$ space, (iv) there exist nonregular languages accepted by two-way deterministic pushdown automata within strong $\log\log n$ space, and, (v) there exist unary nonregular languages accepted by two-way one-counter automata using quantum and classical states with middle $\log n$ space and bounded error. 
Zuzana Bedn\'arov\'a, Viliam Geffert, Klaus Reinhardt, and Abuzer   Yakaryilmaz
cs.FL cs.CC quant-ph