> They say that they keep CO2 in liquid form at room temperature, then turn it into gas, and grab the energy so released.

To evaporate something, you need to give it energy (heat). The energy flux through the dome walls is not huge, so CO2 boils away slowly.

> - To turn the gas into the liquid, they need to compress it; this will produce large amounts of heat. It will need large radiators to dissipate (and lose), or some kind of storage to be reused when expanding the gas. What could that be?

Well, you have this giant heatsink called "the atmosphere".

> - How can the whole thing have a 75% round-trip efficiency, if they use turbines that only have about 40% efficiency in thermal power plants?

A quirk of thermodynamics. CO2 is not the _hot_ part, it's the _cold_ part of the cycle.

To explain a bit more, if you confine CO2 and let it boil at room temperature, it will get up to around 70 atmospheres of pressure. You then allow it to expand through a turbine. This will actually _cool_ it to below the room temperature, I don't have exact calculations, but it looks like the outlet temperature will be at subzero temperatures.

This "bonus cold" can be re-used to improve the efficiency of storage or for other purposes.