Hi salsabeel, I'd say "boom (in)" or "surge (in)" (singular noun) even though the Russian word is plural. Various technical terms like spike, flare, upsurge, burst, splash, spurt and so on don't work with "progress".
If we represent 'the technical progress' as an incline line or as a moderately raising curve, "всплески прогресса" would look like short-term peak raises over this line (wavelike fluctuations). However, I wouldn't use such a wording myself.
In Russian, it would be most common to say "всплеск активности" in a context related to science. "Всплеск" may work well with "прогресс в отношениях", meaning temporary improvements in a relationship. For the OP, I'd translate it as "breakthrough," albeit that may sound more pretentious than "всплеск".