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We explore the cryptographic power of arbitrary shared physical resources.The most general such resource is access to a fresh entangled quantum state atthe outset of each protocol execution. We call this the Common ReferenceQuantum State (CRQS) model, in analogy to the well-known Common ReferenceString (CRS). The CRQS model is a natural generalization of the CRS model butappears to be more powerful: in the two-party setting, a CRQS can sometimesexhibit properties associated with a Random Oracle queried once by measuring amaximally entangled state in one of many mutually unbiased bases. We formalizethis notion as a Weak One-Time Random Oracle (WOTRO), where we only ask of the$m$--bit output to have some randomness when conditioned on the $n$--bit input.  We show that when $n-m\in\omega(\lg n)$, any protocol for WOTRO in the CRQSmodel can be attacked by an (inefficient) adversary. Moreover, our adversary isefficiently simulatable, which rules out the possibility of proving thecomputational security of a scheme by a black-box reduction to a cryptographicgame assumption. On the other hand, we introduce a non-game quantum assumptionfor hash functions that implies WOTRO in the CRQ\$ model (where the CRQSconsists only of EPR pairs). We first build a statistically secure WOTROprotocol where $m=n$, then hash the output.  The impossibility of WOTRO has the following consequences. First, we show theblack-box impossibility of a quantum Fiat-Shamir transform, extending theimpossibility result of Bitansky et al. (TCC '13) to the CRQS model. Second, weshow a black-box impossibility result for a strenghtened version of quantumlightning (Zhandry, Eurocrypt '19) where quantum bolts have an additionalparameter that cannot be changed without generating new bolts.

Fiat-Shamir for Proofs Lacks a Proof Even in the Presence of Shared Entanglement