A computing system processes memory transactions for parallel processing of multiple threads of execution provides execution of multiple atomic instruction groups (AIGs) on multiple systems to support a single large transaction that requires operations on multiple threads of execution and/or on multiple systems connected by a network. The support provides a Transaction Table in memory and fast detection of potential conflicts between multiple transactions. Special instructions may mark the boundaries of a transaction and identify memory locations applicable to a transaction. A ‘private to transaction’ (PTRAN) tag, directly addressable as part of the main data storage memory location, enables a quick detection of potential conflicts with other transactions that are concurrently executing on another thread. The tag indicates whether (or not) a data entry in memory is part of a speculative memory state of an uncommitted transaction that is currently active in the system.