  We study the stability of Hall MHD with strong magnetic fields in which
Landau quantization of electrons is important. We find that the strong-field
Hall modes can be destabilized by the dependence of the differential magnetic
susceptibility on magnetic field strength. This instability is studied using
linear perturbation theory, and is found to have typical growth time of order
$\lesssim 10^3$ yrs, with the growth time decreasing as a function of
wavelength of the perturbation. The instability is self-limiting, turning off
following a period of local field growth by a few percent of the initial value.
Finite temperature is also shown to limit the instability, with sufficiently
high temperatures eliminating it altogether. Alfv\'{e}n waves can show similar
unstable behaviour on shorter timescales. We find that Ohmic heating due to the
large fields developed via the instability and magnetic domain formation is not
large enough to account for observed magnetar surface temperatures. However,
Ohmic heating is enhanced by the oscillatory differential magnetic
susceptibility of Landau-quantized electrons, which could be important to
magneto-thermal simulations of neutron star crusts.
