Both the molten silicon system and the liquid air system work as heat engines when they’re delivering energy.
The maximum possible thermal efficiency of a heat engine depends on the temperature range over which it works, according to the formula:
Thermal Efficiency = (T2 – T1) / T2
where T2 is the absolute temperature of the heat source, and T1 is the absolute temperature of the heat sink.
Silicon melts at 1414°C, which is 1687°K*, and that is T2 for the silicon system. Ambient temperature in the environment varies, but is typically around 0°C - 30°C – which is 273°K - 303°K but if you start dumping heat to the environment, you warm it up locally, and T1 for your system will be higher than the ambient temperature. How much higher depends how much you warm the environment locally, which depends on how much heat you dump, and the thermal resistance of the system you use to dump the heat – conventional power stations use cooling towers to reduce the thermal resistance, but T1 remains well above ambient temperature, and in addition energy is used to pump the cooling water up the cooling towers.
T2 for the liquid air system is lower than ambient, because of the thermal resistance of the environment as a heat source. The heat sink in this case is the liquid air, at −194°C which is 79°K.
The liquid air system works as a heat pump when it’s storing energy. (Theoretically, the molten silicon system could, too, but in practice it probably simply uses resistive heating.)
A heat pump can pump more heat than the energy it uses to pump it. The maximum coefficient of performance is simply the inverse of the maximum efficiency of the heat engine over the same temperature range. That is, a perfect heat pump / heat engine combination could theoretically deliver exactly as much energy as had originally been stored in it. Of course in practice we can’t achieve anywhere near that round-trip efficiency.
A substantial part (but only a part) of the loss in these systems is the thermal resistance of the connection to the environment:
Providing a direct connection from (1) to (3) eliminates two sets of thermal resistance. (A direct connection from (2) to (4) would eliminate two more, if a heat pump were used to melt the silicon.)
* °K is the measure of absolute temperature – that is, 0°K is absolute zero.