Possible to check if $a \in \mathrm{QR}_n$? It is possible to check $a \in \mathrm{QR}_p \text{ iff } a^{(p-1)/2} \equiv 1\ (\bmod\ p)$ if $p$ is a prime.

$n$ is a large RSA modulus. Is it also possible to check if $a \in \mathrm{QR}_n$ if the factorization of $n$ is unknown?
It is possible to check $a \in \mathrm{QR}_p \text{ iff } a^{(p-1)/2} \equiv 1\ (\bmod\ p)$ if $p$ is a prime.


Yes, you have already written down the check, which is known as Euler's criterion.


  $n$ is a large RSA modulus. Is it also possible to check if $a \in \mathrm{QR}_n$ if the factorization of $n$ is unknown?


You can use the Jacobi symbol if the factorization is unknown to find out if $a\in \mathrm{QNR}_n$, i.e., if the Jacobi symbol is $-1$. 

But if the Jacobi symbol yields $+1$, then you can either have a quadratic residue or a pseudosquare, i.e., a quadratic non-residue yielding Jacobi symbol $+1$.

Distinguishing pseudosquares from quadratic residues modulo an RSA modulus if the factorization is unknown is known as the quadratic residuosity problem which is believed to be a hard problem and is the basis for some cryptographic constructions such as the Goldwasser-Micali encryption scheme. 

For generic algorithms this problem can be shown to be equivalent to factoring.