  Given a verifier circuit for a problem in QMA, we show how to exponentially
amplify the gap between its acceptance probabilities in the `yes' and `no'
cases, with a method that is quadratically faster than the procedure given by
Marriott and Watrous. Our construction is natively quantum, based on the
analogy of a product of two reflections and a quantum walk. Second, in some
special cases we show how to amplify the acceptance probability for good
witnesses to 1, making a step towards the proof that QMA with one-sided error
is equal to QMA. Finally, we simplify the filter-state method to search for QMA
witnesses by Poulin and Wocjan.
