The first thing to understand about time series correlations is that they are not necessarily significant – a lot of the time they are simply due to chance.
For instance, if you were to measure the heights of two individuals every year from birth till age 18, you would find a strong positive correlation in height growth – both individuals will grow in height if they are in good health.
However, this long-term growth trend is largely a function of time and does not provide any meaningful insights into the relationship between the two series. Simply because one individual is growing in height, this does not mean that it is in any way influencing the second person’s growth in height – they are entirely separate events.
Cointegration is a way of testing if a correlation between two time series is indeed significant or simply due to chance. A cointegrated pair is one where the individual series are non-stationary (do not have a constant mean, variance, or autocorrelation), but a linear combination of these series makes them stationary (they do have a constant mean, variance, and autocorrelation).
As a starting point, it would make sense to test for stationarity using a test such as Dickey-Fuller across different time periods, e.g. 1990-1995, 1996-2000, etc. If you find non-stationarity, first difference the series and then test for stationarity again. If you see evidence of cointegration across these time periods, then this is a significant indication that the two market indices are significantly correlated.