  We show that the precision of an angular measurement or rotation (e.g., on
the orientation of a qubit or spin state) is limited by fundamental constraints
arising from quantum mechanics and general relativity (gravitational collapse).
The limiting precision is $r^{-1}$ in Planck units, where $r$ is the physical
extent of the (possibly macroscopic) device used to manipulate the spin state.
This fundamental limitation means that spin states $S_1$ and $S_2$ cannot be
experimentally distinguished from each other if they differ by a sufficiently
small rotation. Experiments cannot exclude the possibility that the space of
quantum state vectors (i.e., Hilbert space) is fundamentally discrete, rather
than continuous. We discuss the implications for finitism: does physics require
infinity or a continuum?
