Quantum speedup without entanglement

Question

Is there an instance of a quantum algorithm that is faster than its classical counterpart, but doesn't use entanglement, only superposition?

Answer 1

No. Without entanglement we can always write the system as the product state of individual qbits, and those qbits are just a pair of complex numbers. We can thus simulate the quantum system on a classical computer in polynomial time & space, and would not gain any benefit from execution on a quantum computer.

There are methods of analysis by which a quantum computer outperforms a classical computer without entanglement such as query complexity (number of times the black box function is queried) in the Deutsch Oracle problem, but these do not translate into "real world" speedups and are mostly of interest to complexity theorists. When we talk about quantum speedups in the real world, it is usually a physical quantum computer compared to a classically-simulated quantum computer.