title={IP = PSPACE},
  author={Adi Shamir},
  journal={J. ACM},
  • A. Shamir
  • Published 1 October 1992
  • Mathematics
  • J. ACM
In this paper, it is proven that when both randomization and interaction are allowed, the proofs that can be verified in polynomial time are exactly those proofs that can be generated with polynomial space. 
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In [4], it was claimed that the amount of communication in an interactive protocol for QBFormulaSAT can be bounded by a polynomial in the number of variables in the input formula. However, the proof
Two-Message Quantum Interactive Proofs Are in PSPACE
It is proved that QIP(2), the class of problems having two-message quantum interactive proof systems, is a subset of PSPACE by means of an efficient parallel algorithm, based on the matrix multiplicative weights update method, for approximately solving a certain class of semidefinite programs.
IP = SPACE: simplified proof
A slightly simplified version of Shamir's proof is presented, using degree reductions instead of simple QBFs, to prove that PH is contained in IP.
Computational Complexity and Mathematical Proofs
How the major computational complexity classes, P, NP and PSPACE, capture different computational properties of mathematical proofs and reveal new quantitative aspects of mathematics are discussed.
Interactive Proof Systems and Alternating Time-Space Complexity
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We show that given a satisfiable instance of the 2-to-1 Label Cover problem, it is NP-hard to find a ( 23 24 + )-satisfying assignment.
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More on BPP and the Polynomial-Time Hierarchy
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  • Mathematics, Computer Science
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  • 1996
Progress on Polynomial Identity Testing - II
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  • 2013
The area of algebraic complexity theory is surveyed, with the focus being on the problem of polynomial identity testing (PIT), and the key ideas are discussed.
On Random Oracle Separations


E-mail and the unexpected power of interaction
  • L. Babai
  • Computer Science
    Proceedings Fifth Annual Structure in Complexity Theory Conference
  • 1990
The LFKN protocol, interactive proofs, complexity classes, relativized separation, arithmetization of Boolean formulas, program verification, multiple provers, circuit reductions and publishable
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This technique is used to prove that every language in the polynomial-time hierarchy has an interactive proof system and played a pivotal role in the recent proofs that IP = PSPACE and MIP = NEXP.
The Complexity of Computing the Permanent
  • L. Valiant
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 1979
Proofs that yield nothing but their validity and a methodology of cryptographic protocol design
This paper demonstrates the generality and wide applicability of zero-knowledge proofs, a notion introduced by Goldwasser, Micali and Rackoff that efficiently demonstrate membership in the language without conveying any additional knowledge.
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Permission to copy without fee all or part of this material is granted provided that the copies arc not made or distributed for direct commercial advantage. rhe ACM copyright notice and the title of
Every Prime has a Succinct Certificate
  • V. Pratt
  • Mathematics, Computer Science
    SIAM J. Comput.
  • 1975
It remains an open problem whether a prime n can be recognized in only $\log _2^\alpha n$ operations of a Turing machine for any fixed $\alpha $.
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  • Seinosuke Toda
  • Computer Science
    30th Annual Symposium on Foundations of Computer Science
  • 1989
It follows that neither PP nor (+)P is a subset of or equivalent to PH unless PH collapses to a finite level, strong evidence that both classes are strictly harder than PH.
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A model in which a computationally bounded verifier consults with an oracle in the presence of malicious faults on the communication lines, and it is shown that a deterministic polynomial time verifier can test membership in any language in P-space, but cannottest membership in languages not in P -space, even if he is allowed to toss random coins in private.