Hi QJB.
Glad you came back and posted again.
The efficiency of any system is:
what you get out / what you put in
Carnot's Theorem tells us that in reality even with an idealised engine (no friction losses etc) efficiency is determined by the difference between the temperature at which heat enters the engine and the ambient temperature of the surrounding environment. Since we cannot really hope to change the ambient temperature of the environment the best we can do is improve the temperature at which the heat enters the engine. In aircraft this is done by increasing the temperature at which the fuel air mixture is ignited.
Yup, exactly what I've been saying from the start - this is the
thermodynamic reason why higher compression engines are more efficient. Glad it helped you answer the question from your OP.
Obviously how the engine is designed, the operating RPM range, type of fuel etc. etc. are all things which are important for designing an engine for a certain application.