That is really a policy question. The constitution actually provides fewer protections to non-citizens than it does to citizens. It's not unusual for countries to do that. The question is what are bottom line protections that should extend to everyone. Those questions are not always answered the way people might expect, and there has been a lot of movement in that area since 9/11, but there was a long history before that as well. I'm not by any means an expert in the area, but I'm quite sure a basic history of immigration in the United States would give you an idea of the general parameters. Check the web or your local library.
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It depends on the context of your question. In some instances, the Constitution specifically protects or references only citizens. In other situations it protects all persons. In a criminal context, for example, the Fifth Amendment reads as follows:
No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offense to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation.
In criminal cases, unless you are charged with terrorism against the United States, you have certain rights regardless of your citizenship.
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In most cases a non-citizen will have the same rights as a citizen, such as the 5th amendment right against self-incrimination, first amendment right of freedom of speech, 6th amendment right to counsel, 14th amendment right to due process of law, etc. In a few situations the non-citizen will have fewer rights. This can be changed either by getting the constitution changed (a long, complex procedure that has only been done a few times in over 200 years), or by getting legislation passed to increase the rights of non-citizens.
Whether it "should" be done is not really a legal question, but a policy question.
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