  In this article we construct Laurent polynomial Landau-Ginzburg models for
cominuscule homogeneous spaces. These Laurent polynomial potentials are defined
on a particular algebraic torus inside the Lie-theoretic mirror model
constructed for arbitrary homogeneous spaces in arXiv:math/0511124. The Laurent
polynomial takes a similar shape to the one given in arXiv:alg-geom/9603021 for
projective complete intersections, i.e. it is the sum of the toric coordinates
plus a quantum term. We also give a general enumeration method for the summands
in the quantum term of the potential in terms of the quiver introduced in
arXiv:math/0607492, associated to the Langlands dual homogeneous space. This
enumeration method generalizes the use of Young diagrams for Grassmannians and
Lagrangian Grassmannians and can be defined type-independently. The obtained
Laurent polynomials coincide with the results obtained so far in
arXiv:1404.4844 and arXiv:1304.4958 for quadrics and Lagrangian Grassmannians.
We also obtain new Laurent polynomial Landau-Ginzburg models for orthogonal
Grassmannians, the Cayley plane and the Freudenthal variety.
