The widespread use of machine learning has raised the question of quantum supremacy for supervised learning as compared to quantum computational advantage. In fact, a recent work shows that computational and learning advantage are, in general, not equivalent, i.e., the additional information provided by a training set can reduce the hardness of some problems. This paper investigates under which

... [Show full abstract] conditions they are found to be equivalent or, at least, highly related. The existence of efficient algorithms to generate training sets emerges as the cornerstone of such conditions. These results are applied to prove that there is a quantum speed-up for some learning tasks based on the prime factorization problem, assuming the classical intractability of this problem.