  In multi-parameter quantum metrology, the resource of entanglement can lead
to an increase in efficiency of the estimation process. Entanglement can be
used in the state preparation stage, or the measurement stage, or both, to
harness this advantage; here we focus on the role of entangling measurements.
Specifically, entangling or collective measurements over multiple identical
copies of a probe state are known to be superior to measuring each probe
individually, but the extent of this improvement is an open problem. It is also
known that such entangling measurements, though resource-intensive, are
required to attain the ultimate limits in multi-parameter quantum metrology and
quantum information processing tasks. In this work we investigate the maximum
precision improvement that collective quantum measurements can offer over
individual measurements for estimating parameters of qudit states, calling this
the 'collective quantum enhancement'. We show that, whereas the maximum
enhancement can, in principle, be a factor of $n$ for estimating $n$
parameters, this bound is not tight for large $n$. Instead, our results prove
an enhancement linear in dimension of the qudit is possible using collective
measurements and lead us to conjecture that this is the maximum collective
quantum enhancement in any local estimation scenario.
