  Thermodynamic Uncertainty Relations (TURs) are relations that establish lower
bounds for the relative fluctuations of thermodynamic quantities in terms of
the statistics of the associated entropy production. In this work we derive a
family of TURs that explores higher order moments of the entropy production and
is valid in any situation a Fluctuation Theorem holds. The resulting bound
holds in both classical and quantum regimes and can always be saturated. These
TURs are shown in action for a two level system weakly coupled to a bath
undergoing a non time-symmetric drive, where we can use the Tasaki-Crooks
fluctuation theorem. Finally, we draw a connection between our TURs and the
existence of correlations between the entropy production and the thermodynamic
quantity under consideration.
