  We explain the mechanism of the quantum speed-up - quantum algorithms
requiring fewer computation steps than their classical equivalent - for a
family of algorithms. Bob chooses a function and gives to Alice the black box
that computes it. Alice, without knowing Bob's choice, should find a character
of the function (e. g. its period) by computing its value for different
arguments. There is naturally correlation between Bob's choice and the solution
found by Alice. We show that, in quantum algorithms, this correlation becomes
quantum. This highlights an overlooked measurement problem: sharing between two
measurements the determination of correlated (thus redundant) measurement
outcomes. Solving this problem explains the speed-up. All is like Alice, by
reading the solution at the end of the algorithm, contributed to the initial
choice of Bob, for half of it in quantum superposition for all the possible
ways of taking this half. This contribution, back evolved to before running the
algorithm, where Bob's choice is located, becomes Alice knowing in advance half
of this choice. The quantum algorithm is the quantum superposition of all the
possible ways of taking half of Bob's choice and, given the advanced knowledge
of it, classically computing the missing half. This yields a speed-up with
respect to the classical case where, initially, Bob's choice is completely
unknown to Alice.
