  We compare several topologies on the Morse boundary $\partial_M Y$ of a
$\mathrm{CAT(0)}$ cube complex $Y$. In particular, we show that the two
topologies introduced by Cashen and Mackay are not equal in general and provide
a new description of one of them in the language of cube complexes. As a
corollary, we obtain a new approach to tackle the question whether the visual
topology induces a quasi-isometry-invariant topology on the Morse boundary.
This leads to an obstruction to quasi-isometry-invariance in terms of the
behaviour of geodesics under quasi-isometries.
