Braverman, Finkelberg, and Nakajima define the $K$-theoretic Coulomb branch of a 3$d$ $\mathcal{N}=4$ SUSY gauge theory as the affine variety $\mathcal{M}_{G,N}$ arising as the equivariant $K$-theory of certain moduli space $\mathcal{R}_{G,N}$, labelled by the complex reductive group $G$ and its complex representation $N$. It was conjectured by Gaiotto, that (quantized) $K$-theoretic Coulomb branches bear the structure of (quantum) cluster varieties. I will outline a proof of this conjecture for quiver gauge theories, and show how the cluster structure allows to count the BPS states (aka DT-invariants) of the theory. Time permitting, I will also show how the above cluster structure relates to positive and Gelfand-Tsetlin representations of quantum groups. This talk is based on joint works with Gus Schrader.