Inclusion Criteria:

          -  Calculated creatinine clearance (using Cockcroft-Gault equation) >= 30 mL/min

          -  Absolute neutrophil count >= 1000/mL

          -  Untransfused platelet count >= 75000/mL

          -  Hemoglobin >= 8.0 g/dL

          -  Total bilirubin =< 1.5 x the upper limit of the normal range (ULN)

          -  Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =< 3 x ULN

          -  Patients with relapsed multiple myeloma who have already received one or more standard
             treatment regimens

          -  Measurable disease of multiple myeloma as defined by at least ONE of the following:

               -  Serum monoclonal protein >= 1.0 g/dL

               -  >= 200 mg of monoclonal protein in the urine on 24 hour electrophoresis

               -  Serum immunoglobulin free light chain >= 10 mg/dL AND abnormal serum
                  immunoglobulin kappa to lambda free light chain ratio

          -  Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1, 2

          -  Patients should be proteasome inhibitor naïve (including bortezomib and carfilzomib)
             OR have received less than 6 cycles of therapy with a bortezomib or carfilzomib
             containing regimen and were not refractory to the bortezomib or carfilzomib based
             regimen (less than a PR or progression on or within 60 days of discontinuation)

          -  Provide informed written consent

          -  Negative pregnancy test done =< 7 days prior to registration, for women of
             childbearing potential only

          -  Willing to return to consenting Mayo Clinic institution for follow-up during the
             Active Monitoring Phase of the study; Note: during the Active Monitoring Phase of a
             study (i.e., active treatment and observation), participants must be willing to return
             to the consenting institution for follow-up

          -  Recovered (i.e., < grade 1 toxicity) from the reversible effects of prior
             antineoplastic therapy

        Exclusion Criteria:

          -  Recent prior chemotherapy:

               -  Alkylators (e.g. melphalan, cyclophosphamide) =< 14 days prior to registration

               -  Anthracyclines =< 14 days prior to registration

               -  High dose corticosteroids, immune modulatory drugs (thalidomide or lenalidomide)
                  =< 7 days prior to registration

          -  Prior therapy with any proteasome inhibitor other than bortezomib or carfilzomib

          -  Concomitant high dose corticosteroids other than what is part of treatment protocol
             (concurrent use of corticosteroids); EXCEPTION: patients may be on chronic steroids
             (maximum dose 20 mg/day prednisone equivalent) if they are being given for disorders
             other than myeloma, i.e., adrenal insufficiency, rheumatoid arthritis, etc

          -  Diagnosed or treated for another malignancy within 2 years before study enrollment or
             previously diagnosed with another malignancy and have any evidence of residual
             disease; patients with nonmelanoma skin cancer or carcinoma in situ of any type are
             not excluded if they have undergone complete resection

          -  Any of the following:

               -  Pregnant women or women of reproductive ability who are unwilling to use 2
                  effective methods of contraception from the time of signing the informed consent
                  form through 90 days after the last dose of study drug

               -  Nursing women

               -  Men who are unwilling to use a condom (even if they have undergone prior
                  vasectomy) while having intercourse with any women, while taking the drug and for
                  30 days after stopping treatment

          -  Other co-morbidity which would interfere with patient's ability to participate in
             trial, e.g. uncontrolled infection, uncompensated heart or lung disease

          -  Other concurrent chemotherapy, radiotherapy, or any ancillary therapy considered
             investigational; NOTE: bisphosphonates are considered to be supportive care rather
             than therapy, and are thus allowed while on protocol treatment

          -  Patient has >= grade 3 peripheral neuropathy, or grade 2 with pain on clinical
             examination during the screening period

          -  Major surgery within 14 days before study registration

          -  Systemic treatment with strong inhibitors of cytochrome P450, family 1, subfamily A
             (CYP3A) inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin,
             phenobarbital), or use of Ginkgo biloba or St. John's wort within 14 days before the
             first dose of study treatment

          -  Evidence of current uncontrolled cardiovascular conditions, including cardiac
             arrhythmias, congestive heart failure, angina, or myocardial infarction within the
             past 6 months; Note: prior to study entry, any electrocardiogram (ECG) abnormality at
             screening must be documented by the investigator as not medically relevant

          -  Known human immunodeficiency virus (HIV) positive

          -  Known hepatitis B surface antigen-positive status, or known or suspected active
             hepatitis C infection

          -  Any serious medical or psychiatric illness that could, in the investigator's opinion,
             potentially interfere with the completion of treatment according to this protocol

          -  Known allergy to any of the study medications, their analogues or excipients in the
             various formulations

          -  Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral
             absorption or tolerance of MLN9708 including difficulty swallowing

          -  Diarrhea > grade 1, based on the National Cancer Institute (NCI) Common Terminology
             Criteria for Adverse Events (CTCAE) grading, in the absence of antidiarrheals