Financial time series analyses have played an important role in developing some of the fundamental economic theories. However, many of the published analyses of financial time series focus on long-term average behavior of a market, and thus shed little light on the temporal evolution of a market, which from time to time may be interrupted by stock crashes and financial crises. Consequently, in terms of complexity science, it is still unknown whether the market complexity during a stock crash decreases or increases. To answer this question, we have examined the temporal variation of permutation entropy (PE) in Chinese stock markets by computing PE from high-frequency composite indies of two stock markets: the Shanghai Stock Exchange (SSE) and the Shenzhen Stock Exchange (SZSE). We have found that PE decreased significantly in two significant time windows, each encompassing a rapid market rise and then a few gigantic stock crashes. One window started in the middle of 2006, long before the 2008 global financial crisis, and continued up to early 2011. The other window was more recent, started in the middle of 2014, and ended in the middle of 2016. Since both windows were at least one year long, and proceeded stock crashes by at least half a year, the decrease in PE can be invaluable warning signs for regulators and investors alike.

How different are the emerging and the well-developed stock markets in terms of efficiency? To gain insights into this question, we compared an important emerging market, the Chinese stock market, and the largest and the most developed market, the US stock market. Specifically, we computed the Lempel–Ziv complexity (LZ) and the permutation entropy (PE) from two composite stock indices, the Shanghai stock exchange composite index (SSE) and the Dow Jones industrial average (DJIA), for both low-frequency (daily) and high-frequency (minute-to-minute)stock index data. We found that the US market is basically fully random and consistent with efficient market hypothesis (EMH), irrespective of whether low- or high-frequency stock index data are used. The Chinese market is also largely consistent with the EMH when low-frequency data are used. However, a completely different picture emerges when the high-frequency stock index data are used, irrespective of whether the LZ or PE is computed. In particular, the PE decreases substantially in two significant time windows, each encompassing a rapid market rise and then a few gigantic stock crashes. To gain further insights into the causes of the difference in the complexity changes in the two markets, we computed the Hurst parameter H from the high-frequency stock index data of the two markets and examined their temporal variations. We found that in stark contrast with the US market, whose H is always close to 1/2, which indicates fully random behavior, for the Chinese market, H deviates from 1/2 significantly for time scales up to about 10 min within a day, and varies systemically similar to the PE for time scales from about 10 min to a day. This opens the door for large-scale collective behavior to occur in the Chinese market, including herding behavior and large-scale manipulation as a result of inside information.