In the book 'Computational Complexity' by Arora and Barak the following question is posed (exercise 2.20.):

Let REALQUADEQ be the language of all satisfiable sets of quadratic equations over real variables. Show that REALQUADEQ is NP-complete.

I know how to show NP-hardness, but I'm stuck when it comes to proving that this problem is in NP, in particular how to show that we can describe a solution using a polynomial number of bits.

I did some research and found out that over the complex numbers, it remains an open question if the problem is in NP [1]. It also seems closely related to the existential theory of the reals, which again is not known to be in NP.

Thus my question: Is this problem known to be in NP? And if so, could somebody point me in the right direction regarding the proof.

[1] Pascal Koiran, Hilbert’s Nullstellensatz is in the Polynomial Hierarchy, Journal of Complexity 12 (1996), no. 4, pp. 273–286.


1 Answer 1


This problem (usually called QUAD) is in fact complete for the existential theory of the reals, and thus showing NP-membership would be a fantastic and unexpected result. It seems that Arora and Barak intended only to ask for a proof of NP-hardness.

  • $\begingroup$ I came to the same conclusion in the past days, thanks for the confirmation. Indeed, it seems to be a mistake in the book. In section 2.7.4. (where they reference Exercise 2.20) they call the problem NP-complete as well and write regarding membership in NP: "If a solution exists, then that solution serves as a certificate to this effect (of course, we have to show that the solution can be described using a polynomial number of bits, which we omit)." I will inform the authors about this error to avoid further confusion in the future. $\endgroup$
    – mwien
    Commented Jun 22, 2020 at 18:51
  • $\begingroup$ There seem to be some inconsistency there as well, since in the next line Hilbert’s Nullstellensatz is mentioned. However Hilbert’s Nullstellensatz does not hold over the reals. $\endgroup$ Commented Jun 22, 2020 at 23:13

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