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In the present study, we investigate the multifractal nature of along-cadence time series observed by the \textit{Kepler} mission for a sampleof 34 M dwarf stars and the Sun in its active phase. Using the MultifractalDetrending Moving Average algorithm (MFDMA), which enables the detection ofmultifractality in nonstationary time series, we define a set of multifractalindices based on the multifractal spectrum profile as a measure of the level ofstellar magnetic activity. This set of indices is given by the ($A$,$\Delta\alpha$,$C$,$H$)-quartet, where $A$, $\Delta \alpha$ and $C$ are related togeometric features from the multifractal spectrum and the global Hurst exponent$H$ describes the global structure and memorability of time series dynamics. Asa test, we measure these indices and compare them with a magnetic index definedas $S_{ph}$ and verify the degree of correlation among them. First, we applythe Poincar\'e plot method and find a strong correlation between the$\left\langle S_{ph}\right\rangle$ index and one of the descriptors thatemerges from this method. As a result, we find that this index is stronglycorrelated with long-term features of the signal. From the multifractalperspective, the $\left\langle S_{ph}\right\rangle$ index is also stronglylinked to the geometric properties of the multifractal spectrum except for the$H$ index. Furthermore, our results emphasize that the rotation period of starsis scaled by the $H$ index, which is consistent with Skumanich's relationship.Finally, our approach suggests that the $H$ index may be related to theevolution of stellar angular momentum and a star's magnetic properties.

New Suns in the Cosmos IV: the multifractal nature of stellar magnetic activity in \textit{Kepler} cool stars