Observations of two typical contrasting weakly stable and very stable boundary layers from the winter at Dome C station, Antarctica, are used as a benchmark for two centimetre-scale-resolution large-eddy simulations. By taking the Antarctic winter, the effects of the diurnal cycle are eliminated, enabling the study of the long-lived steady stable boundary layer. With its homogeneous, flat snow surface, and extreme stabilities, the location is a natural laboratory for studies on the long-lived stable boundary layer. The two simulations differ only in the imposed geostrophic wind speed, which is identified as the main deciding factor for the resulting regime. In general, a good correspondence is found between the observed and simulated profiles of mean wind speed and temperature. Discrepancies in the temperature profiles are likely due to the exclusion of radiative transfer in the current simulations. The extreme stabilities result in a considerable contrast between the stable boundary layer at the Dome C site and that found at typical mid-latitudes. The boundary-layer height is found to range from approximately 50m to just 5m in the most extreme case. Remarkably, heating of the boundary layer by subsidence may result in thermal equilibrium of the boundary layer in which the associated heating is balanced by the turbulent cooling towards the surface. Using centimetre-scale resolutions, accurate large-eddy simulations of the extreme stabilities encountered in Antarctica appear to be possible. However, future simulations should aim to include radiative transfer and sub-surface heat transport to increase the degree of realism of these types of simulations.