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We develop a bootstrap approach to Euclidean and Lorentzian two-point correlators in quantum mechanical systems. Using positivity constraints together with the Heisenberg equations of motion, the problem can be formulated as a semidefinite program. Solving this problem in its primal form naively requires time discretization and hence non-rigorous. We show that by going to the dual formulation, the Heisenberg equations of motion become "inequalities of motion" on the Lagrange multipliers enforcing the constraints, allowing one to obtain rigorous bounds on two-point correlators from a finite-dimensional semidefinite or polynomial matrix program. We illustrate the method by bootstrapping the Euclidean and Lorentzian two-point correlators in the ungauged one-matrix quantum mechanics in both the ground state and thermal state, from which we extract the spectrum and matrix elements of the low-lying adjoint states. This work is based on 2511.08560 and ongoing work with Henry Lin and Zechuan Zheng.