Query sensitivity of time series under differential privacy
I think you should explain what is $x_i$ in this time series. I would recommend linking the paper. I think without a strong context we could answer wrongly because maybe you could have understood something wrong as it has happened to me sometimes. Some points:

Query output could be $\mathbb{R}$ or $\mathbb{R}^n$, depends what kind of query you are defining and using.

You are right, I suspect they are defining a global sensitivity because maybe is easier to compute. But this depends on the kind of query they are defining on the paper, is a linear query, an only-one query  or a non-linear query