A simple necessary decoherence condition for a set of histories
  Within the decoherent histories formulation of quantum mechanics, we investigate necessary conditions for decoherence of arbitrarily long histories. We prove that fine-grained histories of arbitrary length decohere for all classical initial states if and only if the unitary evolution preserves classicality of states (using a natural formal definition of classicality). We give a counterexample showing that this equivalence does not hold for coarse-grained histories. 
Artur Scherer, Andrei N. Soklakov, Ruediger Schack (Royal Holloway,   University of London)