Can States/Localities Prosecute Federal Election Crimes?

According to Whitehouse.gov "Federal elections are administered by state and local governments, although the specifics of how elections are conducted differ between the states. The Constitution and laws of the United States grant the states wide latitude in how they administer elections."

So, let's take a purely hypothetical scenario where a Presidential candidate commits election fraud. Since the States and localities hold their own jurisdictional authority over federal elections, wouldn't election fraud against the United States necessarily entail fraud against them as well? And could individual States and localities, therefore, prosecute the candidate in addition to whatever crimes Federal authorities might or might not choose to prosecute?