(1) "Our finding implies that there is no a priori theoretical quantum speed-up associated with Grover’s algorithm"
(2) "we show that there is no theoretical quantum advantage unless proven otherwise and quantum advantage has to be decided in a case-by-case manner"
(1) is surprising (at least for me). I took the quadratic speedup of GA as proven.
(2) concedes that GA may be faster for certain quantum oracles but it has to be shown.
Further they completely ignore that when you open up the oracle like they have done, the problem they are considering is really CIRCUIT-SAT, and in this case the grover algorithm yields a 2^{n/2} algorithm whereas the best classical algorithm is 2^n. That the classical algorithm cannot do better that 2^n is the "exponential time hypothesis". I don't think the authors want to claim that they have disproven this hypothesis, since they didn't really. They just showed in some cases, in CIRCUIT-SAT, the problem is easy. This is a fairly benign, "yes...and....", statement.
So I think this is word games where the game the authors has played is to chose the worst words to describe their result. It's a bit sad because the authors are trying to think about the role of entanglement in these algorithms, and where entanglement is low we know that we can efficiently simulate classically these quantum systems.
The first half of the paper, about opening up the oracle is written by definition for people who don't know it, as is the case in any article. Personally knowing about something but which is not published is not a valid or helpful criticism of a published work (or preprint). On the other hand we could not find any publications talking about opening the oracle in the context of attempting to simulate it nor discussing entanglement barriers in the oracle (other than giving unhelpfully general worst-case bounds). The one exception is the following paper by Chamon and Mucciolo https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.10.... If you know of some publications you could point us to, we'd be happy to incorporate them into a later draft of the article and cite them.
"There are many cases where it's already known one doesn't need Grover's algorithm, such as if a problem already has a polynomial-time solution. We have now identified a new set of cases where one doesn't need Grover's, which is where the oracle can be simulated only once by a tensor network (or log(N) times in a "closed" simulation".
So the point of that part of the article is to further delineate when Grover's algorithm is actually needed or not needed. It only applies to real-life problems where one must actually know the circuit.
And thanks for your lecture notes.