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Quantum mechanics brought to us concepts as the Planck length and the Planck time — i.e. the shortest measurable length and the shortest measurable interval of time it makes sense talking about.

By analogy, is there such a thing as a Planck angle, i.e. an angle whose amplitude is so small that no theoretically known improvement in measurement instruments could measure an angle narrower than that?

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I'm not sure your first sentence is right: Planck length and time arise from natural units wherein all the fundamental physical constants are taken to be unity. Thus the notions of Planck Length and Time simply arise from the definition of a particularly convenient system of units.

As for what these units have to do with quantum mechanics and physics in general is answered, for example, by the Plank Length Wikipedia page:

"There is currently no proven physical significance of the Planck length."

Likewise for the Planck time. Some as yet experimentally unvalidated theories ascribe a physical significance to these lengths. In any case, it is widely believed in Physics that future theories - particularly of quantum gravity - will show how hitherto unknown behaviors peculiar to very small length / time intervals arise.

As for whether there is such an analogy for angle, that is just as much in question as for length / time scale significance.

Note, on an unrelated topic, that there is a relationship between angle and quantization: the compactness of the angle space (the compactness of the circle) is what gives rise to the quantization of angular momentum. Angular momentum in physics comes in discrete, countable units, where linear momentum, arising from the noncompact real line domain for position, does not and can take on any value. See This Physics SE question for more details.

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Here's a foolish thought experiment on the subject. Consider now the smallest angle imaginable: take a power of ten and raise it to some negative exponent as huge as your heart's content. Now, imagine the length of the arc formed, given a large enough radius, say, a few billion light-years long. For the smaller the angle may be, with a radius big enough, any angular amount is conceptually significant. Thus, the tiniest rotation possible can only be as measurable as the observation method accuracy order allows it and it is an infinitesimal limit, thus, there shall be no such thing as a "geometrical Planck angle", it is an abstraction, although there may even be [or not, at same time] question considered within a Quantum environment reality.

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