We show that several well-known one-dimensional quantum systems possess a hidden non-local supersymmetry. The simplest example is the open XXZ spin chain with = −1/2. We use the supersymmetry to place lower bounds on the ground-state energy with various boundary conditions. For an odd number of sites in the periodic chain, and with a particular boundary magnetic field in the open chain, we can derive the ground-state energy exactly. The supersymmetry thus explains why it is possible to solve the Bethe equations for the ground state in these cases. We also show that a similar spacetime supersymmetry holds for the t-J model at its integrable ferromagnetic point, where the spacetime supersymmetry and the Hamiltonian it yields coexist with a global u(1|2) graded Lie algebra symmetry. Possible generalizations to other algebras are discussed.At J = ±2t, and the particular chemical potential given in (12), the t-J Hamiltonian has a global u(1|2) symmetry rotating the three states on each site into each other [15]. One can thus think of the J = 2t case as the antiferromagnetic u(1|2) Heisenberg model, and the J = −2t case as the ferromagnetic one. Three of the nine generators of u(1|2) symmetry are fermionic, while the other six are bosonic, so this is a graded Lie algebra. The generators areBecause of the fermionic generators, such a symmetry is often called 'supersymmetry' in condensed matter physics. One should keep in mind that it is different from the spacetime