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The problem of estimating a member of a scalar random signal sequence with quantum-mechanical measurements is considered. The minimum variance linear estimator based on an optimal present quantum measurement and optimal linear processing of past measurements is found. When the average optimal measurement without postprocessing, for a fixed signal, is linear in the random signal and the signal sequence is pairwise Gaussian, the optimal processing separates: the optimal measurement is the same as the optimal measurement without regard to past data, and the past and present data are processed classically. The results are illustrated by considering the estimator of the real amplitude of a laser signal received in a single-mode cavity along with thermal noise; when the random signal sequence satisfies a linear recursion, the estimate can be computed recursively. For a one-step memory signal sequence it is shown that the optimal observable generally differs from the optimal observable disregarding the past; the optimal measurement can be computed recursively.