The three concepts of the title are closely related in statistical mechanics and condensed matter. Order is an organisational principle that gives a material its macroscopic properties.

Equilibration is the most common phenomenon in Nature. There is no other law that is stronger than the one that imposes systems to eventually reach equilibrium.

On the other hand, memory is the property of staying away from equilibrium. If one has to store information, one has to keep a system away from equilibrium for long times.

All these concepts can be illustrated by the functioning of a device that we use every day: the Hard Drive. In this talk, I will tackle the following question: can one build a Quantum Hard Drive?

In this talk, I show that the existence of exotic phases of the matter called topological phases - which do not rely on the symmetry breaking mechanism - provides a mechanism for keeping quantum states alive for a long time, and therefore to build a quantum hard drive. This kind of order relies on a topological pattern of entanglement. I will address the problem of the survival of Topological Order at finite temperature and after a quantum quench. Time permitting, I will discuss some novel routes towards a stable quantum memory, including many-body localization.