  The antiferromagnetic topological insulator (AFTI) is topologically protected
by the combined time-reversal and translational symmetry $\mathcal{T}_c$. In
this paper we investigate the effects of the $s$-wave superconducting pairings
on the multilayers of AFTI, which breaks $\mathcal{T}_c$ symmetry and can
realize quantum anomalous Hall insulator with unit Chern number. For the weakly
coupled pairings, the system corresponds to the topological superconductor
(TSC) with the Chern number $C=\pm 2$. We answer the following questions
whether the local Chern numbers and chiral Majorana edge modes of such a TSC
distribute around the surface layers. By the numerical calculations based on a
theoretic model of AFTI, we find that when the local Chern numbers are always
dominated by the surface layers, the wavefunctions of chiral Majorana edge
modes must not localize on the surface layers and show a smooth crossover from
spatially occupying all layers to only distributing near the surface layers,
similar to the hinge states in a three dimensional second-order topological
phases. The latter phase can be distinguished from the former phase by the
measurements of the local density of state. In addition we also study the
superconducting vortex phase transition in this system and find that the
exchange field in the AFTI not only enlarges the phase space of topological
vortex phase but also enhances its topological stability. These conclusions
will stimulate the investigations on superconducting effects of AFTI and drive
the studies on chiral Majorana edge modes and vortex Majorana zero modes into a
new era.
