It not correlation - bell inequalities show that local variables don't explain the correlations: ie no correlated variable exists because states are decided at measurement (or wavefunction collapse) time. This is why entanglement is interesting.
But entanglement itself seems very not-weird to me once you buy into those other things.
One such group of postulates that chemists like to use can be found at http://vergil.chemistry.gatech.edu/notes/quantrev/node20.htm... . You can probably beat the elegance of those if you're willing to step further from what's actually used in calculations, but that's not what chemists like to do.
Here's a better shot at being axiomatic and elegant, but it's a lot less clear (from a lay-perspective) how they relate to reality: https://en.wikipedia.org/wiki/Dirac%E2%80%93von_Neumann_axio...
Quantum Theory From Five Reasonable Axioms