  Shor's factoring algorithm provides a super-polynomial speed-up over all
known classical factoring algorithms. Here, we address the question of which
quantum properties fuel this advantage. We investigate a sequential variant of
Shor's algorithm with a fixed overall structure and identify the role of
coherence for this algorithm quantitatively. We analyze this protocol in the
framework of dynamical resource theories, which capture the resource character
of operations that can create and detect coherence. This allows us to derive a
lower and an upper bound on the success probability of the protocol, which
depend on rigorously defined measures of coherence as a dynamical resource. We
compare these bounds with the classical limit of the protocol and conclude that
within the fixed structure that we consider, coherence is the quantum resource
that determines its performance by bounding the success probability from below
and above. Therefore, we shine new light on the fundamental role of coherence
in quantum computation.
