In this paper, we firstly propose a novel construction of $16$-quadrature amplitude modulation (QAM) near-complementary sequences with low peak-to-mean envelope power ratio (PMEPR) in orthogonal frequency division multiplexing (OFDM) systems. The proposed $16$-QAM near-complementary sequences can be constructed by utilizing novel nonlinear offsets, where the length of the sequences is $n=2^m$... The family size of the newly constructed $16$-QAM near-complementary sequences is $8\times (\frac{m! }{2})\times 4^{m+1}$, and the PMEPR of these sequences is proven to satisfy ${\textrm{PMEPR}}\leq 2.4$. Thus, the proposed construction can generate a number of $16$-QAM near-complementary sequences with low PMEPR, resulting in the improvement of the code rate in OFDM systems. Furthermore, we also propose a novel construction of $64$-QAM near-complementary sequences with low PMEPR, which is the first proven construction of $64$-QAM near-complementary sequences. The PMEPRs of two types of the proposed $64$-QAM near-complementary sequences are proven to satisfy that ${\textrm{PMEPR}}\leq 3.62$ or ${\textrm{PMEPR}}\leq 2.48$, respectively. The family size of the newly constructed $64$-QAM near-complementary sequences is $64\times (\frac{m! }{2})\times 4^{m+1}$. (read more)