  We show that our Universe lives in a topological and non-perturbative vacuum
state full of a large amount of hidden quantum hairs, the hairons. We will
discuss and elaborate on theoretical evidences that the quantum hairs are
related to the gravitational topological winding number in vacuo. Thus, hairons
are originated from topological degrees of freedom, holographically stored in
the de Sitter area. The hierarchy of the Planck scale over the Cosmological
Constant (CC) is understood as an effect of a Topological Memory intrinsically
stored in the space-time geometry. Any UV quantum destabilizations of the CC
are re-interpreted as Topological Phase Transitions, related to the
desapparence of a large ensamble of topological hairs. This process is
entropically suppressed, as a tunneling probability from the N- to the
0-states. Therefore, the tiny CC in our Universe is a manifestation of the rich
topological structure of the space-time. In this portrait, a tiny neutrino mass
can be generated by quantum gravity anomalies and accommodated into a large
N-vacuum state. We will re-interpret the CC stabilization from the point of
view of Topological Quantum Computing. An exponential degeneracy of topological
hairs non-locally protects the space-time memory from quantum fluctuations as
in Topological Quantum Computers.
