  The melting of the corner of a crystal is a classical, real-world,
non-equilibrium statistical mechanics problem which has shown several
connections with other branches of physics and mathematics. For a perfect,
classical crystal in two and three dimensions the solution is known: the
crystal melts reaching a certain asymptotic shape, which keeps expanding
ballistically. In this paper, we move onto the quantum realm and show that the
presence of quenched disorder slows down severely the melting process.
Nevertheless, we show that there is no many-body localization transition, which
could impede the crystal to be completely eroded. We prove such claim both by a
perturbative argument, using the forward approximation, and via numerical
simulations. At the same time we show how, despite the lack of localization,
the erosion dynamics is slowed from ballistic to logarithmic, therefore pushing
the complete melting of the crystal to extremely long timescales.
