  Knowledge extraction, typically studied in the classical setting, is at the
heart of several cryptographic protocols. We introduce the notion of secure
quantum extraction protocols. A secure quantum extraction protocol for an NP
relation $\mathcal{R}$ is a classical interactive protocol between a sender and
a receiver, where the sender gets the instance $z$ and a witness $w$, while the
receiver only gets the instance $z$. For any efficient quantum adversarial
sender (who follows the protocol but can choose its own randomness), there
exists a quantum extractor that can extract a witness $w'$ such that $(z,w')
\in \mathcal{R}$ while a malicious receiver should not be able to output any
valid witness. We study and construct two types of secure quantum extraction
protocols.
  (1) Quantum extraction protocols secure against quantum malicious receivers
based on quantum fully homomorphic encryption satisfying some mild properties
and quantum hardness of learning with errors. In this construction, we
introduce a non black box technique in the quantum setting. All previous
extraction techniques in the quantum setting were solely based on quantum
rewinding.
  (2) Quantum extraction protocols secure against classical malicious receivers
based on quantum hardness of learning with errors.
  As an application, based on the quantum hardness of learning with errors, we
present a construction of constant round quantum zero-knowledge argument
systems for NP that guarantee security even against quantum malicious
verifiers; however, our soundness only holds against classical probabilistic
polynomial time adversaries. Prior to our work, such protocols were known
based, additionally, on the assumptions of decisional Diffie-Hellman (or other
cryptographic assumptions that do not hold against polynomial time quantum
algorithms).
