  Why do we not experience a violation of macroscopic realism in every-day
life? Normally, no violation can be seen either because of decoherence or the
restriction of coarse-grained measurements, transforming the time evolution of
any quantum state into a classical time evolution of a statistical mixture. We
find the sufficient condition for these classical evolutions for spin systems
under coarse-grained measurements. Then we demonstrate that there exist
"non-classical" Hamiltonians whose time evolution cannot be understood
classically, although at every instant of time the quantum spin state appears
as a classical mixture. We suggest that such Hamiltonians are unlikely to be
realized in nature because of their high computational complexity.
