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This could be a really naive question (and honestly I just don't want to dig through ArXiV review papers on lattice QCD), but my question is simple:

Why exactly is Lattice QCD done in $\mathbb{R}^4$ and not Minkowski space?.

I'm a physics graduate student, so answers on this this level are preferred.

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If you use Minkowski space, you will have an oscillating factor $\exp(iS_M)$ in the path integral. This is difficult to deal with numerically. It is often called the 'sign problem.' In Euclidean space you have a real exponential $\exp(-S_E)$ which you can deal with using things like the Metropolis algorithm. Even in Euclidean space there can be imaginary terms in the action arising from things like the vacuum angle, and these need special methods to treat.

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