Progress by the Lattice Hadron Physics Collaboration in determining the baryon and meson resonance spectrum of QCD using Monte Carlo methods with space-time lattices is described. The extraction of excited-state energies necessitates the evaluation of correlation matrices of sets of operators, and the importance of extended three-quark operators to capture both the radial and orbital structures of baryons is emphasized. The use of both quark-field smearing and link-field smearing in the operators is essential for reducing the couplings of the operators to the high-frequency modes and for reducing statistical noise in the correlators. The extraction of nine energy levels in a given symmetry channel is demonstrated, and identifying the continuum spin quantum numbers of the levels is discussed.
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