 
Summary: Quantum CopyProtection and Quantum Money
Scott Aaronson #
Abstract
Forty years ago, Wiesner proposed using quantum
states to create money that is physically impossible to
counterfeit, something that cannot be done in the clas
sical world. However, Wiesner's scheme required a
central bank to verify the money, and the question of
whether there can be unclonable quantum money that
anyone can verify has remained open since. One can
also ask a related question, which seems to be new:
can quantum states be used as copyprotected programs,
which let the user evaluate some function f , but not
create more programs for f?
This paper tackles both questions using the arsenal
of modern computational complexity. Our main result
is that there exist quantum oracles relative to which
publiclyverifiable quantum money is possible, and any
family of functions that cannot be e#ciently learned
from its inputoutput behavior can be quantumly copy
