Understanding and quantifying entanglement in quantum systems is a central theme in quantum information science. On one hand quantum entanglement is a valuable resource that enables novel computation and communication. On the other hand, the fact that some quantum systems have bounded entanglement accounts for the success of computational methods in finding ground states and simulating dynamics. I will give an overview of recent work in understanding the complexity of ground states of Hamiltonians both from the perspective of computational complexity and from the perspective of quantifying entanglement. I will focus in particular on one dimensional systems and show that even in the presence of symmetries such as translational invariance ground states of one dimensional systems can exhibit a high degree of entanglement.
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