The Satake category of perverse sheaves on the affine Grassmannian $Gr_G$ plays a fundamental role in geometric representation theory, providing a geometric realization of the representation category of the Langlands dual group. Less understood is the category of perverse coherent sheaves on $Gr_G$, introduced by Bezrukavnikov-Finkelberg-Mirkovic, which unlike the usual Satake category is neither semisimple nor symmetric. We show that the coherent Satake category of $GL_n$ categorifies a quantum cluster algebra, which on a purely computational level exhibits an infinite collection of simple objects whose relations in the Grothendieck ring may be described in elementary combinatorial terms. This result also fills in mathematically part of a physical picture arising from the study of $4d$ $N=2$ gauge theory. This is joint work with Sabin Cautis.