Is quadratic nonresiduosity in $\textbf{NP}$?
When the modulus is a prime $p$, you can compute quadratic residuosity in polynomial time using the Legendre symbol: $x$ is a quadratic residue mod $p$ iff $(x|p)=1$ or $0$.
When the modulus is a prime power $p^\alpha$, then you can do the same: I believe $x$ is a quadratic residue mod $p^\alpha$ iff $(x|p)=1$ or $0$.  I believe this can be proven using Hensel lifting.
When the modulus is a number $n$ with known prime factorization $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$, you can use the Chinese remainder theorem: $x$ is a quadratic residue mod $n$ iff it is a quadratic residue mod $p_i$ for all prime factors $p_i$ of $n$.
This means that quadratic nonresiduosity mod $n$ can be computed in polynomial time when the factorization of $n$ is known or given.  This also means that, as far as I can see, the language QNR should be in NP: a natural certificate is a prime $p$ that divides $n$, and such that $(a|p)=-1$.  The verifier can verify that $p$ is prime in polynomial time and compute the Legendre symbol $(a|p)$ in polynomial time.  At least, that's how it seems to me.
I don't know why Arora-Barak say that QNR is not known to be NP.  This makes me suspect that I may have misunderstood something, as they know way more about computational complexity than I do.