Understanding which quantum computations genuinely require quantum resources is a central question in quantum computing. Lie-algebraic simulation approaches this problem in the Heisenberg picture by exploiting the algebraic structure of quantum dynamics. When the evolution of relevant observables closes within an efficiently accessible polynomial-dimensional operator space, quantum mean values can be computed classically without simulating the full quantum state. In this talk, I will introduce the underlying framework and present recent advances extending its scope beyond free-fermionic and Gaussian dynamics. These include constructive simulation methods for structured qubit circuits and new exact simulation regimes for interacting bosonic systems. I will illustrate their practical applications through numerical studies of quantum many-body dynamics. Together, these results provide new tools for studying and verifying quantum systems and help sharpen the classical baseline against which claims of quantum advantage must be assessed.