A conjecture formalizing this notion was defined by Freedman and Hastings : called NLTS - it stipulates the existence of locally-defined quantum systems that retain long-range entanglement even at high temperatures. Such a conjecture does not only present a necessary condition for quantum PCP, but also poses a fundamental question on the nature of entanglement itself. To this date, no such systems were found, and moreover, it became evident that even embedding local Hamiltonians on robust, albeit "non-physical" topologies, namely expanders, does not guarantee entanglement robustness.
In this study, refute the intuition that entanglement is inherently fragile: we show that locally-defined quantum systems can, in fact, retain long-range entanglement at high temperatures. To do this, we construct an explicit family of 7-local Hamiltonians, and prove that for such local Hamiltonians ANY low-energy state is hard to even approximately simulate by low-depth quantum circuits of depth o(log(n)). In particular, this resolves the NLTS conjecture in the affirmative, and suggests the existence of quantum systems whose low-energy states are not only highly-entangled but also "usefully"-entangled, in the computational-theoretic sense.