Lattice QCD calculations of scattering phaseshifts and resonance parameters in the two-body sector are becoming precision studies.
Early calculations employed L\"uscher's formula for extracting these quantities at lowest order.
As the calculations become more ambitious, higher-order relations are required.
In this study we derive higher-order quantization conditions and introduce a method to transparently cross-check our results.
This is an important step given the involved derivations of these formulae.
We derive quantization conditions up to $\ell=5$ partial waves in both cubic and elongated geometries, and for states with zero and non-zero total momentum.
All 45 quantization conditions we include here (22 in cubic box, 23 in elongated box) pass our cross-check test.