Recent results in computing excited-state energies and meson-meson scattering phase shifts in lattice QCD are presented. A stochastic method of treating the low-lying modes of quark propagation that exploits Laplacian Heaviside quark-field smearing makes such studies possible now on large 32^3 x 256 and 48^3 x 128 lattices at near physical pion masses. Levels are identified using a variety of probe interpolating operators, which include both single-hadron and a large number of two-hadron operators.
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