In climate sciences or finance, it is common for an extreme event to trigger a sequence of high records in a short period. Furthermore, precipitation measures and stock records are heavy-tailed thus extreme value theory is typically used to plan for societal and economic risk. However, the assessment of time dependence is not systematic, even in the stationary framework. We consider stationary regularly varying time series. First, we review classical methods to address the time dependences of extremes based on the identification of short periods with consecutive exceedances of a high level. In this case, the extremal index gives a summary of the clustering effect. Second, we generalize this notion considering short periods, or blocks, with lp−norm above a high threshold and derive large deviation principles of blocks. Our main goal is to promote the choice p < ∞, rather than the classical one for p = ∞, where the bias is more difficult to control. We show the theory developed can be used to improve inference of functionals acting on extreme blocks. For example, the extremal index has an interpretation in this way. It can also be applied to compute accurate confidence intervals of extreme return levels.