State succinctness of two-way finite automata with quantum and classical states

{\it Two-way quantum automata with quantum and classical states} (2QCFA) were introduced by Ambainis and Watrous in 2002. In this paper we study state succinctness of 2QCFA. For any $m\in {\mathbb{Z}}^+$ and any $\epsilon<1/2$, we show that: {enumerate} there is a promise problem $A^{eq}(m)$ which can be solved by a 2QCFA with one-sided error $\epsilon$ in a polynomial expected running time with a constant number (that depends neither on $m$ nor on $\varepsilon$) of quantum states and $\mathbf{O}(\log{\frac{1}{\epsilon})}$ classical states, whereas the sizes of the corresponding {\it deterministic finite automata} (DFA), {\it two-way nondeterministic finite automata} (2NFA) and polynomial expected running time {\it two-way probabilistic finite automata} (2PFA) are at least $2m+2$, $\sqrt{\log{m}}$, and $\sqrt[3]{(\log m)/b}$, respectively; there exists a language $L^{{twin}(m)={wcw|} w\in{a,b}^*}$ over the alphabet $\Sigma={a,b,c}$ which can be recognized by a 2QCFA with one-sided error $\epsilon$ in an exponential expected running time with a constant number of quantum states and $\mathbf{O}(\log{\frac{1}{\epsilon})}$ classical states, whereas the sizes of the corresponding DFA, 2NFA and polynomial expected running time 2PFA are at least $2^m$, $\sqrt{m}$, and $\sqrt[3]{m/b}$, respectively; {enumerate} where $b$ is a constant.