Usually Pearson's correlation coefficient is used in functional imaging time-series, which can be computed over the entire time period or in sliding-windows of various lengths, for example see:

Article Can sliding-window correlations reveal dynamic functional co...

Hi Titas,

I can recommend a fairly new statistic conceptualized in classical test theory (CTT) that computes a 'difference-score' over time and yields an interpretable effect size (D^2ptc; See, file attached).

If your question is about whether to account for within cluster correlation in time series data, I would suggest using the intraclass correlation coefficient (ICC). This is certainly appropriate for mixed effects models. The ICC can help you decide whether accounting for clustering improves modeling of the outcome, as a large ICC may indicate that there is less variability between observations within clusters, and consequently, greater variation in observations between clusters. Depending on your modeling approach, you may also be able to test for differences in model fit between nested models that account for clustering and that do not account for clustering.

I am not sure I am getting your question right, sounds like you need to do clustering of many time series and you would like some correlation measure to use as a clustering statistics. It's probably impossible to define a scalar for the purpose, however you could use the spectrum (estimated as the pooled periodogram of the time series) or maybe some of its possible transformation. One example could be the variance profile (https://amstat.tandfonline.com/doi/abs/10.1080/01621459.2012.682832?casa_token=dQ2rEOs68PYAAAAA:egrZpVP8gM7wZdAF48KuL2lSP9xsg7wgAMftNgKWDYoXoeEPUtsDB4vHaLl06yRMYpkvEnriDwmM) or the Article The Generalised Autocovariance Function