  In this paper, we focus on determining the equivalence for {\it 1-way quantum
finite automata with control language} (CL-1QFAs) defined by Bertoni et al and
{\it measure-many 1-way quantum finite automata} (MM-1QFAs) introduced by
Kondacs and Watrous. More specifically, we obtain that: \begin{enumerate}
  \item[(i)] Two CL-1QFAs ${\cal A}_1$ and ${\cal A}_2$ with control languages
(regular languages) ${\cal L}_1$ and ${\cal L}_2$, respectively, are equivalent
if and only if they are $(c_1n_1^2+c_2n_2^2-1)$-equivalent. Furthermore, if
${\cal L}_1$ and ${\cal L}_2$ are given in the form of DFAs, with $m_1$ and
$m_2$ states, respectively, then there exists a polynomial-time algorithm
running in time $O ((m_1n_1^2+m_2n_2^2)^4)$ that takes as input ${\cal A}_1$
and ${\cal A}_2$ and determines whether they are equivalent. \item[(ii)] Two
MM-1QFAs ${\cal A}_1$ and ${\cal A}_2$ with $n_1$ and $n_2$ states,
respectively, are equivalent if and only if they are
$(3n_1^2+3n_2^2-1)$-equivalent. Furthermore, there is a polynomial-time
algorithm running in time $O ((3n_1^2+3n_2^2)^4)$ that takes as input ${\cal
A}_1$ and ${\cal A}_2$ and determines whether ${\cal A}_1$ and ${\cal A}_2$ are
equivalent.
