QIP = PSPACE

QIP is equal to PSPACE.

Statement

Migrated verbatim from quantum-interactive-proofs § Known relationships:

• $\mathbf{QIP}=\mathbf{PSPACE}$ — TODO citation (Jain, Ji, Upadhyay, Watrous 2010). Since $\mathbf{IP}=\mathbf{PSPACE}$ as well, quantum interactive proofs are no more powerful than classical interactive proofs. This is a striking collapse: quantum communication between prover and verifier adds no power to multi-round interactive proofs.

Migrated verbatim from quantum-statistical-zero-knowledge § Known relationships:

• $\mathbf{QSZK}\subseteq \mathbf{QIP}=\mathbf{PSPACE}$: quantum statistical zero-knowledge is contained in quantum interactive proofs, which equals PSPACE.

Notes

source: folklore: the claim carried no citation on the page it was migrated from, and none was invented.

class: unstated: no citing page says which notion of reduction is meant. Recording a class the wiki does not state would add a claim.

class: free because a containment between complexity classes is proved by any argument at all; the reduction-class axis does not discriminate here, and unstated would wrongly suggest the information is missing.

Recorded during migration and not fixed — these are claims about the source text, not changes to it:

• MISSING CITATION: ‘TODO citation (Jain, Ji, Upadhyay, Watrous 2010)’ with no reference page linked.
• TYPING LOSS: the equality reads as a one-way inclusion once split.
• TYPING LOSS (the defect this task targets): the parent’s ’=’ becomes a one-way sub-edge, so QIP = PSPACE would migrate as QIP subset PSPACE and the hard direction (PSPACE subset QIP is the easy one; QIP subset PSPACE is Jain-Ji-Upadhyay-Watrous) is silently dropped.
• Uncited here; the same equality is marked with a TODO citation at content/Complexity/quantum-interactive-proofs.md:22.