The decline of Mars' global magnetic field some 3.8–4.1 billion years ago is thought to reflect the demise of the dynamo that operated in its liquid core. The dynamo was probably powered by planetary cooling and so its termination is intimately tied to the thermochemical evolution and present-day physical state of the Martian core. Bottom-up growth of a solid inner core, the crystallization regime for Earth's core, has been found to produce a long-lived dynamo leading to the suggestion that the Martian core remains entirely liquid to this day. Motivated by the experimentally-determined increase in the Fe–S liquidus temperature with decreasing pressure at Martian core conditions, we investigate whether Mars' core could crystallize from the top down. We focus on the “iron snow” regime, where newly-formed solid consists of pure Fe and is therefore heavier than the liquid. We derive global energy and entropy equations that describe the long-timescale thermal and magnetic history of the core from a general theory for two-phase, two-component liquid mixtures, assuming that the snow zone is in phase equilibrium and that all solid falls out of the layer and remelts at each timestep. Formation of snow zones occurs for a wide range of interior and thermal properties and depends critically on the initial sulfur concentration, ξ 0 . Release of gravitational energy and latent heat during growth of the snow zone do not generate sufficient entropy to restart the dynamo unless the snow zone occupies at least 400 km of the core. Snow zones can be 1.5–2 Gyrs old, though thermal stratification of the uppermost core, not included in our model, likely delays onset. Models that match the available magnetic and geodetic constraints have ξ 0 ≈ 10 % and snow zones that occupy approximately the top 100 km of the present-day Martian core.