  We call quantum security the area of IT security dealing with scenarios where
one or more parties have access to quantum hardware. This encompasses both the
fields of post-quantum cryptography (that is, traditional cryptography
engineered to be resistant against quantum adversaries), and quantum
cryptography (that is, security protocols designed to be natively run on a
quantum infrastructure, such as quantum key distribution).
  In this work, we propose the first systematic classification of quantum
security scenarios, and for each of them we recall the main tools and results,
as well as presenting new ones. We achieve this goal by identifying four
distinct quantum security classes, or domains, each of them encompassing the
security notions and constructions related to a particular scenario. We start
with the class QS0, which is `classical cryptography' (meaning that no quantum
scenario is considered). Regarding post-quantum cryptography, we introduce the
class QS1, where we discuss in detail the problems arising when designing a
classical cryptographic object meant to be resistant against adversaries with
local quantum computing power, and we provide a classification of the possible
quantum security reductions in this scenario when considering provable
security. In respect to hybrid classical-quantum models, in the security class
QS2 we discuss in detail the possible scenarios where these scenarios arise,
and what a correct formalization should be in terms of quantum oracle access.
Finally, in the class QS3 we consider all those cryptographic constructions
designed to run natively on quantum hardware.
  We believe that the framework we introduce in this work will be a valuable
tool for the scientific community in addressing the challenges arising when
formalizing sound constructions and notions of security in the quantum world.
