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Public verification of quantum money has been one of the central objects inquantum cryptography ever since Wiesner's pioneering idea of using quantummechanics to construct banknotes against counterfeiting. So far, we do not knowany publicly-verifiable quantum money scheme that is provably secure fromstandard assumptions.  In this work, we provide both negative and positive results for publiclyverifiable quantum money.  **In the first part, we give a general theorem, showing that a certainnatural class of quantum money schemes from lattices cannot be secure. We usethis theorem to break the recent quantum money scheme of Khesin, Lu, and Shor.  **In the second part, we propose a framework for building quantum money andquantum lightning we call invariant money which abstracts some of the ideas ofquantum money from knots by Farhi et al.(ITCS'12). In addition to formalizingthis framework, we provide concrete hard computational problems looselyinspired by classical knowledge-of-exponent assumptions, whose hardness wouldimply the security of quantum lightning, a strengthening of quantum money wherenot even the bank can duplicate banknotes.  **We discuss potential instantiations of our framework, including an oracleconstruction using cryptographic group actions and instantiations fromrerandomizable functional encryption, isogenies over elliptic curves, andknots.

Another Round of Breaking and Making Quantum Money: How to Not Build It from Lattices, and More