In this talk, I will briefly introduce methods to characterize the irreversibility of time-continuous and weak quantum measurements from a thermodynamic viewpoint. By defining a statistical arrow of time for individual realizations of the measurement process, I will show that measurements are absolutely irreversible, similar to the free expansion of a single gas particle in a box. I will present a cold-atom realization of this idea and conclude by discussing some examples where quantum measurement added noise can be rectified to produce useful work, aid quantum ground state cooling, and fuel the ticks of an autonomous quantum clock.