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  1. Some people are concerned about the outlook of China-US relations.
  2. Some people are concerned about the prospects of China-US relations.

In my dictionary avaliable in Chinese, outlook and prospect have the same meaning(something like what will happen in the future). I am wondering if there are identical and suitable in the above sentence.

My second question is whether the two words are followed by "of" or "for", aka. the prospects/outlook for China-US relations or the prospects/outlook of China-US relations.

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Outlook and prospects are not the same.

"Outlook" generally refers to a point of view about a subject matter. It is not constrained to the future. You may have an outlook on something that has happened in the past as well, or in the present context. So anyone can have an outlook; one person, group of people, an organisation or the whole world.

"Prospect" is used for the likelihood of something happening in future with reference to the context.

With respect to your second question, there is no rule as such on what shall be followed by. Prepositions are used with a word based on the context. So outlook can be followed by 'on', 'of', 'for' depending on what fits better. Refer here for sample examples: https://www.dictionary.com/browse/outlook
For your example, "of" looks to be the right word to be used.

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