  We study the projection on classical spins starting from quantum equilibria.
We show Gibbsianness or quasi-locality of the resulting classical spin system
for a class of gapped quantum systems at low temperatures including quantum
ground states. A consequence of Gibbsianness is the validity of a large
deviation principle in the quantum system which is known and here recovered in
regimes of high temperature or for thermal states in one dimension. On the
other hand we give an example of a quantum ground state with strong nonlocality
in the classical restriction, giving rise to what we call measurement induced
entanglement, and still satisfying a large deviation principle.
