Zeno's arrow paradox is a redefinition of "motion": Quantum physics is not required to deal with Zeno's arrow paradox. The statement of the "paradox" works by invoking the idea of "motion" while only ever considering instants of time, and thus not considering motion as a concept that applies with respect to change over time. All that happens in Zeno's statement of the "paradox", and similar restatements by other authors, is that an assertion about "motion" is made on the basis of the position of a thing at a single instant in time; because a thing occupies a single space at an instant in time, it is "motionless", and since this applies at all instants in time, it is "motionless" at all instants in time --i.e., it is always motionless, and motion is impossible.
Zeno's argument rests on a persuasive definition of "motion" which is different to its real meaning. In Zeno's argument, the concept of "motion" is a property of an object at a single instant in time; it bears no relationship to actual motion, as the concept is used by anyone. Striped of its persuasive definition of "motion", all the argument says is: at any instant in time, everything occupying space is in the same space it is in, and not some other space. (If there is any branch of physics that disputes Zeno's argument, it is not quantum physics, but regular classical mechanics, which quantitatively defines the concept of motion. Simple use of classic physics equations show the ridiculousness of trying to measure motion by position data at a single instant in time. However, even a pre-physics understanding of "motion" is sufficient to refute the argument, so long as you recognise that motion is conceptually describing change in location over time.)
Zeno's argument is a classic case of a philosophical argument that tries to bamboozle people by simply redefining a concept to have a completely different meaning. Since the argument invokes the idea of "motion" but does not ever consider changes of position with respect to time, it is similar to (but not exactly the same as) the stolen concept fallacy. Once "motion" is correctly defined as change in position with over time, it is not correct to say that (at any given instant) an arrow is "motionless" merely because it occupies one space at that point in time. (Whether it is motionless or in motion cannot be determined by its position at a single point in time, but by the rate of change of position with respect to time, taken relative to some other existent used as a reference point.)
A little rant about quantum physics and philosophy: This little rant is not a negative comment on the OP, or his question, but just something that needs to be said in the context of this question. People seem to have this ridiculous fetish for quantum physics, where they act like it solves all the philosophical problems of the world. (And no-one seems to have such a fetish for this as non-physicists.) Theory of mind? Quantum physics will solve it! Zeno's paradox? Quantum physics! Moral laws concerning lifeboat situations? Hell, let's try to apply quantum physics!
This is a dead end --- quantum physics solves exactly zero philosophical problems. It is philosophy that is required to help interpret the data from experiments in quantum physics, to avoid making stupid mystical conclusions from this data. ("Oooh, the cat is both dead and alive - I have transcended the law of non-contradiction!") The vast majority of what is written about quantum physics and philosophy is mystical horse-shit, dressed in fancy pseudo-mathematical verbiage.
With regard to the "uncertainty principle", it is a principle of epistemology, not metaphysics, and it merely circumscribes limits of our ability to measure things. Not only does it have no application to the existence or non-existence of motion, but it is a principle that makes reference to motion, and therefore pre-supposes that motion is a thing.