What a quantum many-body system undergoes unitary evolution punctuated by sporadic measurements, its dynamics can fall into one of two broad classes: (1) "entangling" dynamics, in which the quantum entanglement between two subregions grows linearly in time and ultimately saturates at a value proportional to the subregion volume; and (2) "disentangling" dynamics, in which the entanglement growth rate vanishes and the final entanglement depends only on the area of the subregion boundary. I will discuss our efforts to understand the phase transition between these two classes, including the intuition behind our original prediction of its existence and our later efforts to find a situation where the transition can be solved exactly.