  Let $\mathbb{Z}_p$ be the ring of integers modulo a prime number $p$ where
$p-1$ is a quadratic residue modulo $p$. This paper presents the study of
constacyclic codes over chain rings $\mathcal{R}=\frac{\mathbb{Z}_p[u]}{\langle
u^2\rangle}$ and $\mathcal{S}=\frac{\mathbb{Z}_p[u]}{\langle u^3\rangle}$. We
also study additive constacyclic codes over $\mathcal{R}\mathcal{S}$ and
$\mathbb{Z}_p\mathcal{R}\mathcal{S}$ using the generator polynomials over the
rings $\mathcal{R}$ and $\mathcal{S},$ respectively. Further, by defining Gray
maps on $\mathcal{R}$, $\mathcal{S}$ and $\mathbb{Z}_p\mathcal{R}\mathcal{S},$
we obtain some results on the Gray images of additive codes. Then we give the
weight enumeration and MacWilliams identities corresponding to the additive
codes over $\mathbb{Z}_p\mathcal{R}\mathcal{S}$. Finally, as an application of
the obtained codes, we give quantum codes using the CSS construction.
