Around a dozen election-denying Republican candidates secured their party’s nomination for secretary of state this fall. This is the reality, two years on, that Donald Trump’s election lies have created.

There are three types of election-denying candidates, and each one poses distinct problems for civic integrity. There are the swing-state candidates getting lots of justified attention, running in places like Arizona and Michigan, because their elections could have pivotal, clear national implications in the 2024 presidential campaign.

There are candidates like Chuck Gray in Wyoming, who is all but certain to take office in January, as Democrats didn’t field an opponent. Election-denying candidates in very red states aren’t getting as much attention now, but they likely will come January, when they are officeholders. They will help set policies in their states — many of which will also have Republican-led legislatures and governors — where extremist ideas could become law.

And there are people like Dominic Rapini, Connecticut’s Republican secretary of state nominee, who are running in blue states and unlikely to win. Their campaigns, though, will have critical fallout effects. By virtue of their statewide platforms, even losing candidates can further damage the discourse — in their states and nationally — and increase the risks to our democracy. Election deniers in blue states can uniquely exacerbate Mr. Trump’s undermining of faith in our elections, and they, like their winning counterparts in red states, can set the stage for local election-denying candidates to win now or in the future.

Sixteen days after the 2020 presidential election, Mr. Rapini sent 23 tweets containing the same message to a wide range of figures, from a local radio host to Mr. Trump and his lawyer Rudy Giuliani. “It’s time to admit fraud is real and stop denying it! #fraudeniers,” Mr. Rapini wrote.

A month and a half later, while the Capitol building was under invasion by Mr. Trump’s supporters and more than an hour after Mike Pence, who was then the vice president, was whisked into hiding, Mr. Rapini tweeted at the official account for the Office of Connecticut’s Secretary of the State: “The real COUP has been prosecuted by Democrats with fake Russian collusion theories and wide spread, systematic voter fraud.”

Like Republicans’ recently softened stance on abortion, Mr. Rapini has toned down his rhetoric in recent months. Yet as late as last summer, he remained the board chair of Fight Voter Fraud, a group that claims to “have assembled a ‘silent army’ of volunteer and professional investigators” to look for voter fraud. (He has said he left the board last year.) As recently as this spring, the group was aligned with the attorney Cleta Mitchell’s organization, the Election Integrity Network. Ms. Mitchell was one of the lawyers advising Mr. Trump on the Jan. 2, 2021, call where he asked the Georgia secretary of state to “find” more votes for him there.

With a candidate like Mr. Rapini running in a state like Connecticut, where the last Republican secretary of state left office in 1995 (and the last one before that left office in 1959), it would be easy enough to mistake his nomination as unimportant.

Richard L. Hasen, a law professor at U.C.L.A. and a leading election law expert, cautioned anyone who might ignore such candidacies. “First of all, just running these races politicizes even further the office of secretary of state,” he told me. Additionally, Mr. Hasen said that having a candidate on a statewide ballot making “constant false claims of massive voter fraud can’t help but create more doubt about election integrity in the minds of a lot of people.”

Although President Biden carried Connecticut by over 20 points in 2020, about 715,000 residents voted for Mr. Trump. If national estimates of support for Mr. Trump’s election lies were to apply in Connecticut, that would mean as many as half a million voters don’t trust elections there already. A statewide candidacy by someone so dedicated to pressing unfounded claims — even if unsuccessful — could, at the least, solidify that election skepticism.

It’s crucial to remember what the office actually means: In Connecticut, as in many states across the country, the secretary of state is the lead elections official. As the commissioner of elections, Connecticut’s secretary of state is responsible for administering its election laws and, under federal law, doing the same for federal elections. Mr. Rapini has made clear that he would use the position to focus on “election security,” as he wrote on the anniversary of the Capitol insurrection. This would be a marked change from the former secretary of state, who was elected three times and focused on “making voter registration and casting a ballot more convenient and obstacle free for every eligible Connecticut citizen.”

Stephanie Thomas, a first-term Connecticut state representative and longtime nonprofit fund-raising and strategy consultant, is the Democratic nominee facing Mr. Rapini in November. “As a nonprofit fund-raiser, the adage used to be if you send someone an email three times, they think they know your organization and they’re more likely to give. So we know that repetition can sometimes prove effective, even if the message is incorrect,” she said in an interview last month. “This type of false narrative just chips away at the fabric of the integrity of our elections, and I think that is just as dangerous as someone in a more reliably red state saying the election was stolen.”

One of the people already in these offices who went through the 2020 election and its aftermath agrees. Jena Griswold, the Colorado secretary of state and the chair of the Democratic Association of Secretaries of State, is running for re-election this year. She told me she worries about a wide range of state and local dangers to voting rights, including threats to election workers, excuses for voting restrictions and local election security breaches, because “local officials embrace conspiracies and become security threats themselves.”

The point that harm doesn’t just build up but also trickles down is what most worries Sam Oliker-Friedland, who previously worked on voting rights cases at the Justice Department and is now the executive director at the Institute for Responsive Government.

He said his concern in a state like Connecticut is the effect the candidacy “will have on lower races, especially races for local election officials.” While the secretary of state is the formal head of elections, it’s the registrars of voters and town clerks in Connecticut who are doing “the day-to-day work of running elections.” Many more people are involved in carrying out elections than just the top officials in a given state, and while their roles are important, they’re also much lower profile. Those officials and their races going forward “will be influenced by this discourse coming from the person running for the top elections post in the state,” Mr. Oliker-Friedland warned, adding that the discourse will also affect primary elections in those races going forward and the way those people do their jobs once elected.