(Applied Economic Forecasting using Time Series methods; Ghysels, Marcellino, 2018), in the chapter about forecast evaluation, relates efficiency as "the efficient use of the available information". It goes on arguing that this implies that the optimal h-steps ahead forecast error should be at most correlated of order h-1 and uncorrelated with available information at the time the forecast is made. Finally, if this is not the case, the forecast can be improved upon with a more careful specification.
My question is this: why forecast errors must not be correlated with available information in order for the forecast to be efficient? Is it related to Gauss-Markov? In which way are we using available information efficiently?
Bonus question: How can we improve the forecast?