  Public key quantum money can be seen as a version of the quantum no-cloning
theorem that holds even when the quantum states can be verified by the
adversary. In this work, investigate quantum lightning, a formalization of
"collision-free quantum money" defined by Lutomirski et al. [ICS'10], where
no-cloning holds even when the adversary herself generates the quantum state to
be cloned. We then study quantum money and quantum lightning, showing the
following results:
  - We demonstrate the usefulness of quantum lightning by showing several
potential applications, such as generating random strings with a proof of
entropy, to completely decentralized cryptocurrency without a block-chain,
where transactions is instant and local.
  - We give win-win results for quantum money/lightning, showing that either
signatures/hash functions/commitment schemes meet very strong recently proposed
notions of security, or they yield quantum money or lightning.
  - We construct quantum lightning under the assumed multi-collision resistance
of random degree-2 systems of polynomials.
  - We show that instantiating the quantum money scheme of Aaronson and
Christiano [STOC'12] with indistinguishability obfuscation that is secure
against quantum computers yields a secure quantum money scheme
