  Publicly-verifiable quantum money has been a central and challenging goal in
quantum cryptography. To this day, no constructions exist based on standard
assumptions. In this study, we propose an alternative notion called quantum
cheques (QCs) that is more attainable and technologically feasible. A quantum
cheque can be verified using a public-key but only by a single user.
Specifically, the payer signs the quantum cheque for a particular recipient
using their ID, and the recipient can validate it without the assistance of the
bank, ensuring that the payer cannot assign the same cheque to another user
with a different ID. Unlike quantum money, QCs only necessitate quantum
communication when a cheque is issued by the bank, meaning all payments and
deposits are entirely classical!
  We demonstrate how to construct QCs based on the well-studied
learning-with-errors (LWE) assumption. In the process, we build two novel
primitives which are of independent interest. Firstly, we construct signatures
with publicly-verifiable deletion under LWE. This primitive enables the signing
of a message $m$ such that the recipient can produce a classical string that
publicly proves the inability to reproduce a signature of $m$. We then
demonstrate how this primitive can be used to construct 2-message signature
tokens. This primitive enables the production of a token that can be used to
sign a single bit and then self-destructs. Finally, we show that 2-message
signature tokens can be used to construct QCs.
