[This is akin to a living review, which may improve from time to time. Last edited 2025-10-11.]
This post will summarize the various consistency conditions that can be found discussed in the consistent histories literature. Most of the conditions have gone by different names under different authors (and sometimes even under the same author), so I’ll try to give all the aliases I know; just hover over the footnote markers.
There is an overarching schism in the choice of terminology in the literature between the terms “consistent” and “decoherent”. Manya authors, including Gell-Mann and Hartle, now use the term “decoherent” very loosely and no longer employ “consistent” as an official label for any particular condition (or for the formalism as a whole). Zurek and I believe this is a significant loss in terminology, and we are stubbornly resisting it. In our recent arXiv offering, our rant was thus:
…we emphasize that decoherence is a dynamical physical process predicated on a distinction between system and environment, whereas consistency is a static property of a set of histories, a Hamiltonian, and an initial state. For a given decohering quantum system, there is generally a preferred basis of pointer states [1, 8]. In contrast, the mere requirement of consistency does not distinguish a preferred set of histories which describe classical behavior from any of the many sets with no physical interpretation.
(See also the first footnote on page 3347 of “Classical Equations for Quantum Systems”b which agrees with the importance of this conceptual distinction.)… [continue reading]

