We study two kinds of different problems. One is the multiple independence testing, which can be considered as a kind of generalization of quantum Stein's lemma. We test whether the quantum system is correlated to the classical system or is independent of it. Here, the null hypothesis is composed of states having the quantum system is correlated to the classical system in an arbitrarily varying form. The second problem is the problem of reliable communication over classical-quantum arbitrarily varying multiple access channels (CQ-AVMAC) and establishing its capacity region by giving multiple achievability techniques. We prove that each of these techniques is optimal by proving a converse. Further, for both these techniques, the decoder designed is a \emph{universal} decoder and can achieve any rate pair in the capacity region without time sharing and also these decoders do not depend on the channel and therefore they are universal. Our result covers the case when the channel parameter is continuous, which has not been studied even in the classical case. Further, both these techniques can be easily generalized to the case when there are $T (T>2)$ senders. The design of each of these decoders is based on the study of multiple independence testing. This approach allows us to study the problem of reliable communication over CQ-AVMAC from the point of view of hypothesis testing. Further, we also give a necessary and sufficient condition for the deterministic code capacity region of CQ-AVMAC to be non-empty.