Possibly naive question about quantized space-time

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)?

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