The construction of the future is moving in the direction of environmentally friendly materials and the use of various types of industrial byproducts and wastes. The use of blast furnace slag (BFS) for the production of concrete is one of those alternatives. In this study, pastes and concretes based on high-MgO BFS were alkali activated with 10% by weight sodium carbonate, sodium silicate, and a combination of both. Heat treatment and laboratory curing were applied. The results showed that heat treatment was effective at reducing the drying shrinkage of alkali-activated slag concretes and promoting high early strength. However, the sodium carbonate–activated slag concrete specimens showed a reduction in compressive strength at later ages. All concrete specimens tested exhibited high drying shrinkage; the highest values were for sodium silicate–activated concretes and the lowest were for sodium carbonate–activated concretes. All concretes tested showed very large creep, which was partly related to the small maximum aggregate size (8 mm) and the effects of carbonation. The carbonation depth after 12–24 months was significantly smaller for the heat-treated specimens and for concrete activated with sodium silicate. The carbonation process resulted in a more porous binder matrix, leading to long-term strength loss and increased creep, especially for sodium silicate–activated mixes.