Abstract: ‍ ‍For multiple-input multiple-output （MIMO） radar waveform set， the lower the peak of the cross-correlation functions， the better the orthogonality， and the higher the waveform diversity gain. The studies of the lower bounds on aperiodic cross-correlation functions would help determine the limit value of waveform diversity gain， which is of great significance to the MIMO radar waveform designs and applications. At present， phase coded waveform sets are the most studied MIMO radar waveform sets. And the studies of the lower bounds on the correlation function of the phase coded waveform set are in-depth. This paper reviews the existing researches on the lower bounds， including lower bound on peak side-lobe level of correlation functions， lower bound on integral side-lobe level of correlation functions， lower bound on peak inner products and lower bound on correlation functions of complementary sequence set. Among them， the lower bound on peak side-lobe level of correlation functions is the key metric affecting the MIMO radar waveform diversity gain. There are intrinsic relationships among these bounds， which may help determine the lower bound on peak side-lobe level of correlation functions. Numerical results under different number of waveforms and sequence lengths show that the existing lower bounds are not tight compared with existing phase coded waveform sets with low cross-correlation functions. Particularly， the gap between them is obvious large when the number of waveforms is large. Hence， lower bound on the correlation function of the phase coded waveform set is still an open problem worthy of study.