  Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative
ring and $M$ a graded $R$-module. A proper graded submodule $N$ of $M$ is
called \textit{a graded semiprime submodule} if whenever $r\in h(R)$, $m\in
h(M)$ and $n\in Z^{+}$ with $r^{n}m\in N$, then $rm\in N$. In this paper, we
introduce the concept of graded quasi-semiprime submodule as a generalization
of graded semiprime submodule and show a number of results in this class. We
say that a proper graded submodule $N$ of $M$ is \emph{a graded quasi-semiprime
submodule} if $(N:_{R}M)$ is a graded semiprime ideal of $R$.
