Simulating arbitrary quantum dynamics with classical algorithms is widely believed to be intractable. Yet, by exploiting the structure of certain restricted settings, specialized classical methods can succeed. One particularly promising family - Pauli propagation - recasts simulation in the Pauli basis and often delivers rigorous runtime and error guarantees. At the heart of these guarantees lie two ingredients: the presence of local noise, which dampens long-range interactions, and a degree of “scrambling” in the circuit’s gates. In this talk, we will present our recent results on applying Pauli propagation to both noisy and noiseless circuits. Along the way, we’ll discuss to what extent noise and scrambling influence simulability - and what that tells us about the necessary resources for quantum advantage.