The XOR operation is an important part of most of the existing end-to-end authentication protocols. Will it be part of the post-quantum authentication protocols?.
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You are making a category error. The XOR operation is not a trapdoor function or hardness problem. For an algorithm to be weakened by quantum computers, it has to be easy in one direction and tough in another direction, but easy in both directions if you have a quantum computer. That is not the case with XOR. The concept of quantum resistance does not apply to it. It is a linear function described by a very simple truth table:
\begin{array}{|c|c|c|} \hline \text{Input 1}&\text{Input 2}&\text{Output}\\ \hline 0&0&0\\ 0&1&1\\ 1&0&1\\ 1&1&0\\ \hline \end{array}
That doesn't change just because you have a quantum computer.
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