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Recent work has characterised rigorously what it means for one quantum system to simulate another, and demonstrated the existence of universal Hamiltonians -- simple spin lattice Hamiltonians that can replicate the entire physics of any other quantum many body system. Previous universality results have required proofs involving complicated `chains' of perturbative `gadgets'. In this paper, we derive a significantly simpler and more powerful method of proving universality of Hamiltonians, directly leveraging the ability to encode quantum computation into ground states. This provides new insight into the origins of universal models, and suggests a deep connection between universality and complexity. We apply this new approach to show that there are universal models even in translationally invariant spin chains in 1D. This gives as a corollary a new Hamiltonian complexity result, that the local Hamiltonian problem for translationally-invariant spin chains in one dimension with an exponentially-small promise gap is PSPACE-complete. Finally, we use these new universal models to construct the first known toy model of 2D--1D holographic duality between local Hamiltonians.