Winter's measurement compression theorem stands as one of the most important, yet perhaps less well-known coding theorems in quantum information theory. Not only does it make an illuminative statement about measurement in quantum theory, but it also underlies several other general protocols used for entanglement distillation or local purity distillation. The theorem provides for an asymptotic decomposition of any quantum measurement into an "extrinsic" source of noise, classical noise in the measurement that is independent of the actual outcome, and "intrinsic" quantum noise that can be due in part to the nonorthogonality of quantum states. This decomposition leads to an optimal protocol for a sender to 1) simulate many instances of a quantum measurement acting on many copies of some state and 2) send the outcomes of the measurements to a receiver using as little classical communication as possible while still having a faithful simulation. The protocol assumes that the parties have access to some amount of common randomness, which is a strictly weaker resource than classical communication. In this talk, we provide a full review of Winter's measurement compression theorem, detailing the information processing task, providing examples for understanding it, overviewing Winter's achievability proof, and detailing a new approach to its single-letter converse theorem. We then overview the Devetak-Winter theorem on classical data compression with quantum side information, providing new proofs of the achievability and converse parts of this theorem. From there, we outline a new protocol that we call "measurement compression with quantum side information," a protocol announced in prior work on trade-offs in quantum Shannon theory. This protocol has several applications, including its part in the "classically-assisted state redistribution" protocol, which is the most general protocol on the static side of the quantum information theory tree, and its role in reducing the classical communication cost in a task known as local purity distillation. We finally outline a connection between this protocol and recent work on entropic uncertainty relations in the presence of quantum memory. This is joint with Patrick Hayden, Francesco Buscemi, and Min-Hsiu Hsieh.