  We study the discrete flows generated by the symmetry group of the BPS
quivers for Calabi-Yau geometries describing five dimensional superconformal
quantum field theories on a circle. These flows naturally describe the BPS
particle spectrum of such theories and at the same time generate bilinear
equations of q-difference type which, in the rank one case, are q-Painlev\'e
equations. The solutions of these equations are shown to be given by grand
canonical topological string partition functions which we identify with
$\tau$-functions of the cluster algebra associated to the quiver. We exemplify
our construction in the case corresponding to five dimensional $SU(2)$ pure
Super Yang-Mills and $N_f = 2$ on a circle.
