The concept of crystal collimation exploits the peculiar properties of crystalline materials to deflect highenergy beam particles at angles orders of magnitude larger than what can be achieved with scattering by conventional materials used as primary collimators. This innovative technique is planned to be used to improve the collimation efficiency with heavy-ion beams at the Large Hadron Collider (LHC) and its High-Luminosity upgrade (HL-LHC). The unprecedented proton-equivalent energy range of up to 7 TeV makes this technique particularly challenging due to the small angular acceptance of the crystal channeling process. This paper reviews the recent campaign of measurements performed with 6.8 TeV proton beams at the LHC to characterize the latest generation of crystal collimator devices installed in the machine, in preparation for the deployment in the upcoming heavy-ion runs.