  We consider the quantum difference equation of the Hilbert scheme of points
in $\mathbb{C}^2$. This equation is the K-theoretic generalization of the
quantum differential equation discovered by A. Okounkov and R. Pandharipande.
We obtain two explicit descriptions for the monodromy of these equations -
representation-theoretic and algebro-geometric. In the representation-theoretic
description, the monodromy acts via certain explicit elements in the quantum
toroidal algebra $\frak{gl}_1$. In the algebro-geometric description, the
monodromy features as transition matrices between the stable envelope bases in
the equivariant K-theory and elliptic cohomology. Using the second approach we
identify the monodromy matrices for the differential equation with the
K-theoretic $R$-matrices of cyclic quiver varieties, which appear as
subvarieties in the $3D$-mirror Hilbert scheme. Most of the results in the
paper are illustrated by explicit examples for cases $n=2$ and $n=3$ in the
Appendix.
