  We prove a no-go theorem for storing quantum information in equilibrium
systems. Namely, quantum information cannot be stored in a system with
time-independent Hamiltonian interacting with heat bath of temperature $T>0$
during time that grows with the number of used qubits. We prove it by showing,
that storing quantum information for macroscopic time would imply existence of
perpetuum mobile of the second kind. The general results are illustrated by the
Kitaev model of quantum memory. In contrast, classical information can be
stored in equilibrium states for arbitrary long times. We show how it is
possible via phase-transition type phenomena. Our result shows that there is a
fundamental difference between quantum and classical information in {\it
physical} terms.
