Do you believe states with significant immigrant populations should designate particular seats in their legislatures to represent them?

You could get different opinions on this, but I strenuously disagree with this idea. In the United States, certainly, it would be held to be unconstitutional for a number of reasons, and it opens up the door to a kind of representation that I find to be an anathema to democracy. 

Those in Congress must be chosen "by the people" according to Article I of the Constitution. There is nothing in Article I to suggest...

You could get different opinions on this, but I strenuously disagree with this idea. In the United States, certainly, it would be held to be unconstitutional for a number of reasons, and it opens up the door to a kind of representation that I find to be an anathema to democracy. 

Those in Congress must be chosen "by the people" according to Article I of the Constitution. There is nothing in Article I to suggest "the people" can be divided in this way. The Fourteenth Amendment holds that we cannot abridge any privileges of citizens, and setting aside a seat to represent an immigrant group, I think, would violate that amendment, since representation is a zero-sum game. If a seat is set aside for one group, other groups are being deprived of the opportunity to run for that seat, and others' votes are being diluted. Think back to the even greater flood of immigrants in the past. We had large waves of Irish, German, and Italian immigrants, and had we allotted seats based on immigrant populations, we would have had to provide for each of those, as well as all of the other immigrants who came in large numbers. New York in particular would have had several "designated" seats. The people who were not immigrants would have less representation.

While a democracy should representat all its citizens, if we start dividing them into various ethnic groups for the purpose of giving them their own "voice" in Congress, we are elevating their concerns over other legitimate concerns. In addition, we are setting up a situation in which they are even more likely to be perceived as "the other" and more likely to be resented and discriminated against. This is not the democracy I want. Many people are already quite hateful toward immigrants. Doing this would foster more hatred.  There is a slippery slope here, too; if immigration is a special problem, it could be argued living on a geological fault or in the path of hurricanes is, too. People who have special interests at stake will simply have to suffer along with their lobbyists.