  We investigate the elementary steps at the microscopic level for proton
diffusion in brucite under high pressure, which results from a complex
interplay between two processes: the O-H reorientations motion around the
$\mathbf c$ axis and O-H covalent bond dissociations. First-principle
path-integral molecular dynamics simulations reveal that the increasing
pressure tends to lock the former motion, while, in contrast, it activates the
latter which is mainly triggered by nuclear quantum effects. These two
competing effects therefore give rise to a pressure sweet spot for proton
diffusion within the mineral. In brucite \ce{Mg(OH)2}, proton diffusion reaches
a maximum for pressures close to 70GPa, while the structurally similar
portlandite \ce{Ca(OH)2} never shows proton diffusion within the pressure range
and time scale that we explored. We analyze the different behaviors of brucite
and portlandite, which might constitute two prototypes for other minerals with
the same structure.
