  In this paper, we call a sub-scheme of dimension one in $\mathbb{P}^3$ a
curve. It is well known that the arithmetic genus and the degree of an aCM
curve $D$ in $\mathbb{P}^3$ is computed by the $h$-vector of $D$. However, for
a given curve $D$ in $\mathbb{P}^3$, the two invariants of $D$ do not tell us
whether $D$ is aCM or not. In this paper, we give a classification of curves on
a smooth quintic hypersurface in $\mathbb{P}^3$ which are not aCM.
