How many qubits are needed for quantum computational supremacy?

Quantum computational supremacy arguments, which describe a way for a quantum computer to perform a task that cannot also be done by a classical computer, typically require a computational assumption related to the limitations of classical computation. One common assumption is that the polynomial hierarchy (PH) does not collapse, a stronger version of the statement that P =/= NP, which leads to the conclusion that any classical simulation of certain families of quantum circuits requires time scaling worse than any polynomial in the size of the circuits. However, the asymptotic nature of this conclusion prevents us from calculating exactly how many qubits these quantum circuits must have for their classical simulation to be intractable on modern supercomputers. We refine these quantum computational supremacy arguments and perform such a calculation by imposing fine-grained versions of the non-collapse assumption. Based on these conjectures we conclude that IQP circuits with 180 qubits, QAOA circuits with 360 qubits and boson sampling circuits (i.e. linear optical networks) with 90 photons are large enough for producing samples from their output distributions to be intractable on current technology.