  The alcove model of the first author and A. Postnikov uniformly describes
highest weight crystals of semisimple Lie algebras. We construct a
generalization, called the quantum alcove model. In joint work of the first
author with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, this was shown
to uniformly describe tensor products of column shape Kirillov-Reshetikhin
crystals in all untwisted affine types; moreover, an efficient formula for the
corresponding energy function is available. In the second part of this paper,
we specialize the quantum alcove model to types $A$ and $C$. We give explicit
affine crystal isomorphisms from the specialized quantum alcove model to the
corresponding tensor products of column shape Kirillov-Reshetikhin crystals,
which are realized in terms of Kashiwara-Nakashima columns.
