The techniques developed in this work, such as the new version of the Gowers-Hatami theorem and the analysis of succinct Pauli braiding, have the potential to find applications in various areas of quantum complexity and cryptography:
Quantum Error Correction: The insights gained from the analysis of self-testing protocols and the development of new theorems could be applied to the field of quantum error correction. By understanding how to verify the correctness of quantum operations and states efficiently, researchers can improve the reliability and fault tolerance of quantum computing systems.
Quantum Cryptography: The cryptographic primitives and protocols designed in this work could be adapted for use in quantum key distribution and secure quantum communication. Techniques for verifying quantum states and operations could enhance the security and privacy of quantum cryptographic systems.
Quantum Machine Learning: The concepts of self-testing and verification of quantum operations are crucial in the field of quantum machine learning. By ensuring the correctness of quantum algorithms and models, researchers can build more robust and reliable quantum machine learning systems.