  A measurement of the time-dependent CP-violating asymmetry in
$B_s^0\rightarrow\phi\phi$ decays is presented. Using a sample of proton-proton
collision data corresponding to an integrated luminosity of $5.0$fb$^{-1}$
collected by the $\mbox{LHCb}$ experiment at centre-of-mass energies $\sqrt{s}
= 7$ TeV in 2011, 8 TeV in 2012 and 13 TeV in 2015 and 2016, a signal yield of
around 9000 $B_s^0\rightarrow\phi\phi$ decays is obtained. The CP-violating
phase $\phi_s^{s\bar{s}s}$ is measured to be $-0.073 \pm 0.115$(stat)$\pm
0.027$(syst) rad, under the assumption it is independent on the helicity of the
$\phi\phi$ decay. In addition, the CP-violating phases of the transverse
polarisations under the assumption of CP conservation of the longitudinal phase
are measured. The helicity-independent direct CP-violation parameter is also
measured, and is found to be $|\lambda|=0.99 \pm 0.05 $(stat)$ \pm 0.01
$(syst). In addition, $T$-odd triple-product asymmetries are measured. The
results obtained are consistent with the hypothesis of CP conservation in
$b\rightarrow\bar{s}s\bar{s}$ transitions. Finally, a limit on the branching
fraction of the $B^0\rightarrow \phi\phi$ decay is determined to be
$\mathcal{B}(B^0\rightarrow \phi\phi)<2.7\times 10^{-8}$ at 90% confidence
level.
