There are a lot of different ways of understanding what exactly it is that makes quantum computers more powerful. There isn't one simple clean answer. As a cou

There are a lot of different ways of understanding what exactly it is that makes quantum computers more powerful. There isn't one simple clean answer. As a counterpoint to the "it's the bigger state space" idea, note that quantum computations over a polynomial number of qubits with a polynomial number of steps can be simulated by a classical computer using polynomial space (but not polynomial time). So the quantum state space isn't "intrinsically huge" or beyond-the-reach-of-classcal; at least not in that particular sense. Also note that every quantum state we will ever make will be generated by some tractable number of operations done with a tractable amount of precision. So the reachable in practice states are discrete and compressible into a tractable number of bits [the bits being a description of the operations used to produce them].