  The measurement of physics processes at new energy frontier experiments
requires excellent spatial, time, and energy resolutions to resolve the
structure of collimated high-energy jets. In a future Muon Collider, the
beam-induced backgrounds (BIB) represent the main challenge in the design of
the detectors and of the event reconstruction algorithms. The technology and
the design of the calorimeters should be chosen to reduce the effect of the
BIB, while keeping good physics performance. Several requirements can be
inferred: i) high granularity to reduce the overlap of BIB particles in the
same calorimeter cell; ii) excellent timing (of the order of 100 ps) to reduce
the out-of-time component of the BIB; iii) longitudinal segmentation to
distinguish the signal showers from the fake showers produced by the BIB; iv)
good energy resolution (less than 10%/sqrt(E)) to obtain good physics
performance, as has been already demonstrated for conceptual particle flow
calorimeters. Our proposal consists of a semi-homogeneous electromagnetic
calorimeter based on Lead Fluoride Crystals (PbF2) readout by surface-mount
UV-extended Silicon Photomultipliers (SiPMs): the Crilin calorimeter. In this
paper, the performances of the Crilin calorimeter in the Muon Collider
framework for hadron jets reconstruction have been analyzed. We report the
single components characterizations together with the development of a
small-scale prototype, consisting of 2 layers of 3x3 crystals each.
