In polar regions, where the boundary layer is often stably stratified, atmospheric models produce large biases depending on the boundary-layer parametrizations and the parametrization of the exchange of energy at the surface. This model intercomparison focuses on the very stable stratification encountered over the Antarctic Plateau in 2009. Here, we analyze results from 10 large-eddy-simulation (LES) codes for different spatial resolutions over 24 consecutive hours, and compare them with observations acquired at the Concordia Research Station during summer. This is a challenging exercise for such simulations since they need to reproduce both the 300-m-deep convective boundary layer and the very thin stable boundary layer characterized by a strong vertical temperature gradient (10 K difference over the lowest 20 m) when the sun is low over the horizon. A large variability in surface fluxes among the different models is highlighted. The LES models correctly reproduce the convective boundary layer in terms of mean profiles and turbulent characteristics but display more spread during stable conditions, which is largely reduced by increasing the horizontal and vertical resolutions in additional simulations focusing only on the stable period. This highlights the fact that very fine resolution is needed to represent such conditions. Complementary sensitivity studies are conducted regarding the roughness length, the subgrid-scale turbulence closure as well as the resolution and domain size. While we find little dependence on the surface-flux parametrization, the results indicate a pronounced sensitivity to both the roughness length and the turbulence closure.