# The living PRQS reference

updated 2023-02-23

# Pseudorandom quantum states (PRQS)

## All the papers

Let me start by collecting papers that define PRQS, since there are so many definitions!

• The original paper by JLS18.
• Kretschmer21’s quantum oracle that shows PRU (pseudorandom unitaries, and thus PRQS) exist even when $$BQP = QMA$$. Also shows that $$BQP = PP$$ implies PRQS do not exist.
• AQY21, then AGQY22 talk about pseudorandom function-like states. There is also some related work in MY22a but I can’t parse all the differences.
• KQST22 surprisingly improves this to a classical oracle (even a random oracle) where $$P = NP$$ but 1-PRQS exist. This uses Forrelation.
• BFGVZ22 show that assuming OWF (one-way functions), you can get PRQS with entanglement of any $$\omega(\log n) \le S \le n$$ that are all computationally indistinguishable from each other. They call these pseudoentangled, since there are states that “look entangled” but are not that entangled. (This is tight; if $$S = O(\log n)$$, then a swap test can distinguish PRQS from random states.)
• GIKL22 show that that $$\omega(\log n)$$ $T$-gates are required to prepare computationally pseudorandom quantum states.

### Implications for more complex cryptographic protocols

• GJMZ22 describe what it means to commit to a quantum states.
• MY21 extends Kretschmer’s oracle to show quantum commitments can exist even without OWF.
• HMY22 relate this to quantum public-key encryption.
• AHY23 relate this to quantum money.

• BFV19 show that there is a way to view pseudorandomness as an essential element in AdSCFT dictionary within black holes.
• Brakerski22 relates this to black-hole radiation decoding.

### Other papers I don’t know well enough to summarize:

• Sattath22’s review on uncloneable cryptography.
• BCQ22 and the relationship to EFI pairs
• Doosti22’s thesis on uncloneability.
• ALLZZ20’s approaches for quantum copy-protection.
• MY22b on one-way state generators.
• DMP22 combinatorial approach to quantum pseudorandom functions.
• CX22 have variants of pseudorandom quantum state generators.
• Zhandry21 has a new technique on constructing quantum random functions.
• AKL23 new work on cloning games and uncloneable primitives

## What’s next?

I think it would be nice to define the following:

• Pseudorandom quantum states
• Pseudorandom quantum state generators
• (classical and quantum) pseudorandom functions and generators
• pseudorandom unitaries
• single-copy security and multi-copy security

Also, the papers use slightly different definitions, are they all comparable? At least for a few papers, we can write them out and see how they differ.

Then it would be nice to start listing some open questions. Then we could send it around and ask if people have comments.

More to add? Email me at kmarw@uchicago.edu.