Detecting a series of measurements This is somewhat nonstandard:
My input is a series of values (integers or reals), and my goal is to decide if this series is likely to be a series of measurements of some unknown phenomenon (be it an airplane's altitude, a room's temperature, a daily stock price, etc.). If not a series of measurements - this series can be anything. 
I have no clue how these values were generated or what do they represent (if anything at all). The only thing I do know is that these values are given in some 'natural' order (temporal, spatial or other).
One test I considered:


*

*The average absolute difference between consecutive values is smaller than the average absolute difference between each 2nd element, 3rd element, etc.
avg(abs((aₙ - aₙ₋₁)) < avg(abs((aₙ - aₙ₋₂)) < avg(abs((aₙ - aₙ₋₃)) 
I'll be happy to learn if there are known methods to tackle this problem. Alternatively - any idea would be appreciated.
 A: I don't think you will be very successful, though this may depend on your alternative non-time-series data generating processes (which you say you have no clue about).
Yes, you could follow an approach like the one you outline. It is very similar to fitting an arima model (you could take a look at auto.arima() in the forecast package for R) and declaring a series to be a time series if the resulting model has AR or MA orders >0. Yes, you can do this.
The problem is that correlations between ordered sequences of data can come from different sources than from time orderings. The most common such source may be spatial correlation. Your ordered sequence may be temperatures along a stretch of road, measured simultaneously (so there certainly is no time series aspect to this). Fitting an ARIMA model would reveal strong correlations: temperatures between adjacent measurements are more strongly correlated than between distant measurements. The approach outlined above would believe this is a time series.
