  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.
