In this paper, we prove that the spectrum is continuous on the class of all quasi-$*$-$A(n)$ operators. And we obtain a sufficient condition for a quasi-$*$-$A(n)$ operator to be normal. Finally, we consider the tensor products of quasi-$*$-$A(n)$ operators, giving a necessary and sufficient condition for $T\otimes S$ to be a quasi-$*$-$A(n)$ operator when $T$ and $S$ are both non-zero operators.