  Central exclusive processes can be studied in CMS by combining the
information of the central detector with the Precision Proton Spectrometer
(PPS). PPS detectors, placed symmetrically at more than 200 m from the
interaction point, can detect the scattered protons that survive the
interaction. PPS has taken data at high luminosity while fully integrated in
the CMS experiment. The total amount of collected data corresponds to more than
100 fb$^{-1}$ during the LHC Run 2. PPS consists of 3D silicon tracking
stations as well as timing detectors that measure both the position and
direction of protons and their time-of-flight with high precision. The
detectors are hosted in special movable vacuum chambers, the Roman Pots, which
are placed in the primary vacuum of the LHC beam pipe. The sensors reach a
distance of few mm from the beam. Detectors have to operate in vacuum and must
be able to sustain highly non-uniform irradiation: sensors used in Run 2 have
accumulated an integrated dose with a local peak of $\sim 5 \cdot 10^{15}$
protons/cm$^2$. The timing system is made with high purity scCVD diamond
sensors. A new architecture with two diamond crystals read out in parallel by
the same electronic channel has been used to enhance the detector performance.
In this paper, after a general overview of the PPS detector, we describe the
timing system in detail. The sensor and the dedicated amplification chain are
described, together with the signal digitization technique. Performance of the
detector in Run 2 is reported. Recently the sensors used in Run 2 have been
tested for efficiency and timing performance in a dedicated test beam at DESY.
Preliminary results on radiation damage are reported. Important upgrades of the
timing system are ongoing for the LHC Run 3, with the goal of reaching an
ultimate timing resolution better than 30 ps; they are also discussed here.
