Quasi-complementary sequence sets (QCSSs) have attracted sustained research interests due to the ability to support more users compared to complete complementary codes (CCCs) in multi-carrier code division multiple access (MC-CDMA) systems. In this paper, two classes of aperiodic QCSSs are constructed over the finite rings, which have new parameters in terms of the length of constituent sequences and set size respectively. Most notably, the length of constituent sequences of QCSS derived from the first construction is $p^{n} + 1$ , where p is a prime and n is a positive integer, which has not been covered in the literature. In the second construction, the set size of the proposed binary QCSS is $2^{n+1}$ larger than the existing binary QCSS. In addition, both the proposed aperiodic QCSSs are asymptotically optimal with respect to the correlation lower bound.