  According to Heisenberg's uncertainty relation, there is an ultimate limit to
how precisely we may predict the outcome of position and momentum measurements
on a quantum system. We show that this limit may be violated by an arbitrarily
large factor if one aims, instead, to guess the unknown value of a past
measurement. For experiments on a single quantum system, the precise assignment
of past position and momentum measurement outcomes is accompanied by large
uncertainty about their linear combinations, while we show that entanglement
with an ancillary system permits accurate retrodiction of any such linear
combination. Finally, we show that the outcomes of experiments that jointly
measure multiple linear combinations of position and momentum observables by
means of ancillary probe particles can also be guessed with no formal lower
limit. We present quantitative results for projective measurements and for
generalized measurements where all components are prepared in Gaussian states.
