The stock market is a canonical example of a complex system, in which a large number of interacting agents lead to joint evolution of stock returns and the collective market behavior exhibits emergent properties. However, quantifying complexity in stock market data is a challenging task. In this report, we explore four different measures for characterizing the intrinsic complexity by evaluating the structural relationships between stock returns. The first two measures are based on linear and non-linear co-movement structures (accounting for contemporaneous and Granger causal relationships), the third is based on algorithmic complexity, and the fourth is based on spectral analysis of interacting dynamical systems. Our analysis of a dataset comprising daily prices of a large number of stocks in the complete historical data of NASDAQ (1972-2018) shows that the third and fourth measures are able to identify the greatest global economic downturn in 2007-09 and associated spillovers substantially more accurately than the first two measures. We conclude this report with a discussion of the implications of such quantification methods for risk management in complex systems.