  Computing the distance parameters of a network, including the diameter,
radius, eccentricities and the all-pairs shortest paths (APSP) is a central
problem in distributed computing. This paper investigates he dtistance
parameters in the quantum CONGEST models and establishes almost linear lower
bounds on eccentricities and APSP, which match the classical upper bounds. Our
results imply that there is not quantum speedup for these two problems. In
contrast with the diameter and radius, exchanging quantum messages is able to
save the communication when the networks have low diameters [Le Gall and
Magniez, PODC 2018]. We obtain the lower bounds via a reduction from the
two-way quantum communication complexity of the set intersection [Razborov,
Izvestiya Mathematics 2003].
