Statement of Criterion
A voter giving an additional ranking or positive rating to a less-preferred candidate cannot cause a more-preferred candidate to lose.
Later-no-harm (usually LNH, but sometimes LNHa or LNHarm to avoid confusion with Later-no-help) is satisfied by Instant Runoff Voting, Minmax(pairwise opposition), and Douglas Woodall's Descending Solid Coalitions method. It is trivially satisfied by First-Preference Plurality and Random Ballot, since those methods do not usually regard lower preferences. Virtually every other method fails this criterion.
Later-no-harm is incompatible with the Condorcet criterion.
Later-no-harm guarantees that the method will not use a voter's lower preferences to elect a candidate who that voter likes less than the candidate that would have been elected if this voter had kept his lower preferences a secret.
As a result, voters may feel free to vote their complete ranking of the candidates, which in turn may give the election method more complete information to use to find a winner. There is a tradeoff however, in that this criterion simultaneously minimizes the amount of information that the voting method can use to find a winner.
This criteria is equivalent to the criteria that the system is non-compromising in that it will never elect a compromise (i.e. a Utilitarian winner or Condorcet winner.) This is not universally desired so it cannot be claimed that this criteria is always one which would be desirable to pass. If one wants a system which can elect a compromise winner then it would be desirable to fail this criteria.