  Roger Penrose's Weyl curvature hypothesis states that the Weyl curvature is
small at past singularities, but not at future singularities. We review the
motivations for this conjecture and present estimates for the entropy of our
Universe. We then extend this hypothesis to the quantum regime by demanding
that the initial state of primordial quantum fluctuations be the adiabatic
vacuum in a (quasi-) de~Sitter space. We finally attempt a justification of
this quantum version from a fundamental theory of quantum gravity and speculate
on its consequences in the case of a classically recollapsing universe.
