TITLE: What are the strongest arguments for a genuine quantum computing advantage?
QUESTION [34 upvotes]: Despite having become a fairly mature field with enormous sums of money dumped into research and development, there does not as yet exist a formal proof that quantum computation actually provides an advantage. Having learned a bit about quantum circuits, quantum error correcting codes, and all the related basics of quantum information theory in university, the only arguments that were presented to me for quantum supremacy were:

The existence of Shor's algorithm,
The seeming infeasibility of simulating quantum systems with classical computers,

Neither of which are I find particularly compelling, especially when compared to the evidence for $P \neq NP$ for example.
My question is, what are the current "best" arguments for quantum advantage, assuming an ideal quantum computer (either by direct evidence for an advantage, or against the implications there being no advantage)? Given the lack of a hard proof, I am being deliberately vague in my use of the word "best".
I would hope that given the billions of dollars dumped into the field, that there would be extremely strong conjectures and circumstantial evidence of quantum advantage being real/its negation being false.
Edit: By "ideal quantum computer", I mean a quantum computer in the abstract, similar to how we might consider an abstract turing machine or other classical computer model as opposed to any specific physical architecture with all the related engineering concerns.

REPLY [4 votes]: In addition to Shor's algorithm and to Grover algorithm let me mention two important pieces of information that abstract quantum computer have for some specific goals superior computational powers.

The ability of quantum computers, and even quantum circuits of bounded depth to perform sampling tasks that goes beyond the polynomial hierarchy! (Evidence that ${\bf QuantumSampling \not \subset PH}$.)

The evidence that the class of decision problems that quantum computers can solve goes beyond the polynomial hierarchy! (Evidence that ${\bf BQP \not \subset PH}$)


The first item is a long story related to early works of Terhal and DiVincenzo (2004), Aaronson and Arkhipov (2013) and Bremner, Jozsa, and Shepherd (2011). Related also to Boson Sampling.
The second item (with a weaker type of evidence) is related to a 2018 work by Raz and Tal.
${\bf QuantumSampling}$ (which is one aspect of the power of quantum computers that reflects the power of quantum computers for sampling) may be in a sense stronger than ${\bf BQP}$ (The power of quantum computers for decision problems) since it is a plausible conjecture that ${\bf QuantumSampling \not \subset P^{BQP}}$. (And even ${\bf QuantumSampling} \not \subset {\bf PH^{BQP}}$.)