I'm wrapping my head around how and why quantum computers can provide advantage over classical. A basic and naive argument is that the dimension of the Hilbert space of $n$ qubits grows as $2^n$. However, without exponentially sized circuits of 1- and 2-qubit gates only a tiny fraction of all the Hilbert space can be reached. Another explanation I have encountered is that it is the entanglement (and not simply a dimension of the Hilbert space) that is the resource allowing for quantum speed-ups. Whether this is true or not I have the following question.
Should one expect that an efficient quantum algorithm will produce a highly entangled state at some point? If the entanglement is not maximal, should it be possible in principle to run the algorithm on a fewer number of qubits (outsourcing some part of the computation to a classical computer)?