In mathematical physics, the quantum KZ equations or quantum KnizhnikâÂÂZamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the KnizhnikâÂÂZamolodchikov equations for affine KacâÂÂMoody algebras. They are a consistent system of difference equations satisfied by the N-point functions, the vacuum expectations of products of primary fields. In the limit as the deformation parameter q approaches 1, the N-point functions of the quantum affine algebra tend to those of the affine KacâÂÂMoody algebra and the difference equations become partial differential equations. The quantum KZ equations have been used to study exactly solved models in quantum statistical mechanics.