We investigate sampling procedures that certify that an arbitrary

quantum state on $n$ subsystems is close to an ideal mixed state

$\varphi^{\otimes n}$ for a given reference state $\varphi$, up to

errors on a few positions. This task makes no sense classically: it

would correspond to certifying that a given bitstring was generated

according to some desired probability distribution. However, in the

quantum case, this is possible if one has access to a prover who can

supply a purification of the mixed state.

In this work, we introduce the concept of mixed-state certification, and

we show that a natural sampling protocol offers secure certification in

the presence of a possibly dishonest prover: if the verifier accepts

then he can be almost certain that the state in question has been

correctly prepared, up to a small number of errors.

We then apply this result to two-party quantum coin-tossing. Given that

strong coin tossing is impossible, it is natural to ask ``how close can

we get". This question has been well studied and is nowadays well

understood from the perspective of the bias of individual coin tosses.

We approach and answer this question from a different---and somewhat

orthogonal---perspective, where we do not look at individual coin tosses

but at the global entropy instead. We show how two distrusting parties

can produce a common high-entropy source, where the entropy is an

arbitrarily small fraction below the maximum (except with negligible

probability).