Updated

The Republican presidential candidates head into their next debate Tuesday night in New Hampshire to the backdrop of a new leaderboard.

Several state and national polls released this week for the most part show former Massachusetts Gov. Mitt Romney back in the lead, but with businessman Herman Cain nipping at his heels. At least one new state survey shows Cain pulling even with the front-runner. And the latest polls continue to show Texas Gov. Rick Perry falling back since the last national debate.

In the Granite State, Romney has built up an impressive early lead over his competitors. The Harvard Institute of Politics shows him with a double-digit advantage, leading with 37 percent, followed by Cain at 12 percent. An NBC News-Marist poll out Tuesday showed Romney with 44 percent of likely primary voters in New Hampshire. Cain, and Texas Rep. Ron Paul, were trailing with 13 percent each. Behind them was Perry at 6 percent.

The surveys suggest Romney will be on friendly turf when he appears alongside the rest of the field for a debate Tuesday night at Dartmouth College in Hanover, N.H.

At a debate expected to focus squarely on the economy, Romney will face the task of defending his record as an ex-governor who brings private-sector solutions to government thanks to his experience with investment firm Bain Capital and elsewhere in the business world. Cain, as the ex-CEO of Godfather's Pizza, is trying to claim that mantle, while Perry is working to tout his record as a jobs-creating governor to regain lost momentum.

But outside of New England, Romney's lead is not so robust.

The NBC News-Marist survey showed Romney leading with 23 percent in Iowa. But attracting 20 percent of likely GOP caucus-goers, Cain was close behind. In the next tier were Paul at 11 percent, and Perry and Minnesota Rep. Michele Bachmann with 10 percent each. The Iowa poll of 371 GOP likely caucus-goers was conducted Oct. 3-5, and had a margin of error of 5.1 percentage points. The New Hampshire poll of 691 likely GOP primary voters was conducted during the same period, and had a margin of error of 3.7 percentage points.

Nationally, Perry may be doing a bit better. The latest Gallup poll showed the Texas governor with 15 percent support. Romney held the lead with 20 percent, and Cain once again was close behind with 18 percent.

Another poll out Tuesday showed Cain tied for the lead in Virginia. The Quinnipiac poll showed him and Romney each pulling in 21 percent among Republicans surveyed. That poll, too, showed Perry falling back from a first-place position to third, with 11 percent.

The poll of 345 Republicans was conducted Oct. 5-9, and had a margin of error of 5.3 percentage points.