I beg your pardon in advance if this question is naive.
In Quantum Mechanics, discrete values of measurements occur
only in relation to bound states. This is because of the general solution for the time-independent Shroedinger's equation.
Similarly, in order to quantise space-time, do we need to make space-time compact? How would a compact space-time look like? Would we simply need close it by including $\infty$? Is that physical (infinity doesn't sound very physical to me)?