In this talk we will discuss some aspects of the question: given a group, what is the range of Furstenberg entropy of ergodic stationary actions of it? For the special linear group and its lattices, constraints on this spectrum come from Nevo-Zimmer structure theorems; and entropy values can be realized based on Poisson bundles over stationary random subgroups. The complete description is not yet known for other simple Lie groups. Based on joint work with Jeremie Brieussel.