The coming generation of high-energy particle collider experiments will search for the Higgs boson, supersymmetry, and various other particles. \\ \par Observing any new physics at hadron colliders, such as the Large Hadron Collider at CERN, requires a detailed description of multi-particle scattering events. It is not practical to use traditional quantum field theory methods for all significant processes. \\ \par Moreover, the final formulas turn out to be surprisingly elegant. I present recent advances in computing QCD amplitudes compactly and efficiently. Our methods apply the spinor algebra, twistor geometry, and the analyticity of the S-matrix from a contemporary perspective.