  Cosmological probes are steadily reducing the total neutrino mass window,
resulting in constraints on the neutrino-mass degeneracy as the most
significant outcome. In this work we explore the discovery potential of
cosmological probes to constrain the neutrino hierarchy, and point out some
subtleties that could yield spurious claims of detection. This has an important
implication for next generation of double beta decay experiments, that will be
able to achieve a positive signal in the case of degenerate or inverted
hierarchy of Majorana neutrinos. We find that cosmological experiments that
nearly cover the whole sky could in principle distinguish the neutrino
hierarchy by yielding 'substantial' evidence for one scenario over the another,
via precise measurements of the shape of the matter power spectrum from large
scale structure and weak gravitational lensing.
